THE PARTICLE NATURE OF LIGHT and SPECTROSCOPY

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STEFAN-BOLTZMANN LAW

Integrate (add up) a blackbody spectrum and find that the
FLUX, or ENERGY/TIME/AREA is given by

F = sigma T^4

sigma = 5.67 x 10^{-8} W m ^{-2} s ^{-1}

POWER = FLUX x AREA, or, for a sphere (of AREA = 4 pi R^2)

L = 4 pi sigma R^2 T^4

For example, double T and L rises 16 times.

But double R and L rises 4 times.

Raise T 1% and L rises about 4%.

Raise R 1% and L rises about 2%.

Since T can be measured by Wien's Law and L can be obtained
from the star's brightness and distance% (we'll discuss later)
this formula lets astronomers find the SIZES of stars!

Example: T_1 = 500 K, R_1 = 4000 km; T_2 = 250 K, R_2 = 8000 km

L_1/L_2 = (4 pi sigma R_1^2 T_1^4)/(4 pi sigma R_2^2 T_2^4)

= (R_1/R_2)^2 (T_1/T_2)^4 = (4000 km/8000 km)^2 (500 K/250 K)^4

= (1/2)^2 (2)^4 = (1/4) 16 = 4

OR, Planet 1 has 4 times the luminosity of Planet 2

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LIGHT AS PARTICLES

ELECTROMAGNETIC ENERGY IS CARRIED BY PHOTONS:
SINGLE QUANTA OF LIGHT.

E = h nu = h c / lambda

h = 6.63 x 10^{-34} J.s = 6.63 x 10^{-27} erg.s
is PLANCK's CONSTANT.
(Along with c, the speed of light; e, the charge on an
electron (or proton) and G (Newton's constant of gravity),
h is one of the FUNDAMENTAL CONSTANTS of NATURE.)

These PHOTONS can equally well explain

REFLECTION,

REFRACTION,

TRANSMISSION and

ABSORPTION
as can the Wave picture,

BUT they can't explain

INTERFERENCE and

DIFFRACTION.

On the other hand the WAVE picture can't explain:

The PHOTOELECTRIC EFFECT

and SPECTRAL LINES

while the PARTICLE part of the duality in Quantum Mechanics CAN!

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PHOTOELECTRIC EFFECT

Electrons can be expelled from many materials if light shines upon them.

If the wavelength is TOO LONG (low frequency) nothing happens,
EVEN IF the INTENSITY of the light is HIGH.

Above a CRITICAL FREQUENCY the emitted electrons have a maximum
energy (or velocity) that RISES with the FREQUENCY.

E_e = h nu - h nu_{crit}

Increasing the INTENSITY of light above the critical frequency inceases only
the number of ejected electrons, but NOT their energies.

Einstein pointed out that the wave theory could not explain this,
while quanta of energy, with E = h nu could.

The wave theory predicted that even red light, if intense
enough, would eject electrons -- but this never happened.

The wave theory also said that as the blue light was made brighter,
faster electrons would emerge: instead only more of them came out, but their
maximum kinetic (motion) energy was a function ONLY of the light's FREQUENCY.

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SPECTRAL LINES

By the mid 19th century chemists noticed specific
colors of light coming from particular gases.

Careful measurements indicated each element or
compound produced a UNIQUE SET of EMISSION LINES:
equivalent to FINGERPRINTS identifying the element .

Spectra of the SUN and other STARS showed
emission at most frequencies, but distinct dark bands,
or ABSORPTION LINES, were also detected.

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KIRCHHOFF'S LAWS

Gustav Kirchhoff summarized observations and experiments
in the following EMPIRICAL LAWS of SPECTRA:

A CONTINUOUS (blackbody) SPECTRUM arises from a
SOLID, LIQUID or DENSE GAS.

An EMISSION LINE SPECTRUM arises from a LOW DENSITY HOT GAS.

ABSORPTION LINES SUPERPOSED ON A CONTINUOUS SPECTRUM
arise from a LOW DENSITY
GAS INTERPOSED BETWEEN a
CONTINUUM SOURCE and the OBSERVER.

The wavelengths of BOTH EMISSION AND ABSORPTION LINES
are IDENTICAL for a GIVEN ELEMENT.

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ORIGIN OF SPECTRAL LINES

Those in the radio, mm, IR, visible, UV and most X-ray are due to
QUANTUM TRANSITIONS BY ELECTRONS IN ATOMS and MOLECULES

(Gamma-rays usually come from quantum transitions in the nuclei of atoms.)

ABSORPTION LINES ARISE FROM PHOTONS BEING ABSORBED
BY ATOMS AND EXCITING ELECTRONS TO HIGHER LEVELS.

EMISSION LINES ARISE FROM ELECTRONS DROPPING DOWN
TO LOWER ENERGY LEVELS, EMITTING PHOTONS.

E_2 - E_1 = h nu

Is the equation of CONSERVATION OF ENERGY
FOR PHOTO-EXCITATION or PHOTOEMISSION.
In denser gases frequent collisions between atoms shift the observed
wavelengths (Doppler effect) and smear out the lines.

Once the density is high enough, the spectral lines blur
into a CONTINUUM SPECTRUM.

ELECTRONS can be EXCITED THROUGH:

PHOTO-EXCITATION (PHOTO-ABSORPTION), or

COLLISIONAL EXCITATION (atom collides with another atom or electron)

Here conservation of energy can be expressed as:

E_1 + KE_1 = E_2 + KE_2,

with E the electronic potential energy of the atom and KE the kinetic
energy of the colliding atom or electron.

ELECTRONS can be DE-EXCITED THROUGH:

SPONTANEOUS (PHOTO-DE-EXCITATION) or PHOTO-EMISSION;

COLLISIONAL DE-EXCITATION (no photon out);

STIMULATED PHOTOEMISSION, (really requires 3 energy levels
a photon ``reminds'' an electron to drop to the middle level
after another source of energy pumped many electrons the high level
amplifying the original photon via a "chain reaction"
these photons are in phase or COHERENT (constructive interference)
--- yields LASERS and MASERS.
(Acronyms for: Light [or Microwave]] Amplification through
Stimulated Emission of Radiation)

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SPECTRAL LINES GIVE US INFORMATION ON:

Temperature

Composition

Abundances

Pressure (higher yields broader lines)

Turbulence (higher yields broader lines)

Velocity (towards or away -- see Doppler effect)

Rotation (faster yields broader lines -- see Doppler effect)

Magnetic fields

SO THEY ARE THE MOST IMPORTANT THING FOR MOST OF ASTRONOMY

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DOPPLER EFFECT

An observed wavelength or frequency will differ from the emitted one if there
is a relative motion between the emitter and the observer.

RECESSION ---> the OBSERVED lambda IS LONGER,
or FREQUENCY IS LOWER --- REDSHIFT

APPROACH ---> the OBSERVED lambda IS SHORTER,
or FREQUENCY IS HIGHER --- BLUESHIFT

Delta lambda/lambda = (lambda_obs - lambda_em) / (lambda_em) = v_radial / c

Example: lambda_em = 400.000 nm, lambda_obs = 400.005 nm
What is the velocity of the star?
Delta lambda = 400.005 nm - 400.000 nm = 0.005 nm = 5 x 10^-3 nm
Then v = c (Delta lambda/lambda) = (3.0 x 10^5 km/s) x (5 x 10^-3 nm/4.00 x 10^2 nm)
= (3.0 x 10^5 km/s) x (1.25 x 10^-5) = 3.8 x 10^0 km/s
or the star is moving 3.8 km/s AWAY from us.

We can much more easily HEAR the Doppler effect than SEE it.

WHY?

The speed of sound in air is a little more than 300 m/s (or 1000 ft/s)
while the speed of light in air is 300,000,000 m/s or nearly 1,000,000 times more!

A car travelling 62 mph (or 100 km/h) is moving roughly 30 m/s (really 27.8 m/s)
or a little less than 10% of the speed of sound. You can hear a pitch change of 10% very easily.

But the same car is travelling less than 10^-7 of the speed of light
-- that shift of a 500 nm visible spectral line would be only 0.00005 nm
-- way too small to see and extraordinarily hard to measure: only a few instruments
around the world can do this; they are used to find planets around OTHER stars.

LIGHTNING AND THUNDER

You see lightning; the thunderclap comes later.

Lightning is seen at the speed of light;

Thunder is heard at the speed of sound.

Since sound travels roughly 300 m/s or 1000 ft/s, if the sound arrives
about 3 seconds later the bolt was about 1 km away; 5 sec later, about a
mile away; 0 sec later -- you may be dead from a lightning strike!

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