Spectroscopy: The Interaction of Light with Matter

Astronomers depend heavily on spectroscopy for learning about the velocities and compositions of sources. We will look at the hydrogen atom in some detail because is both the simplest atom and also the most abundant in the Universe.

Some preliminaries:

The formation of spectral lines that can serve as "fingerprints" for atoms results from the fact that electrons can orbit atomic nuclei only at certain distances from the nuclei. These allowed orbits are the result of quantum mechanical behavior -- not just photons have a both wave and particle behavior, all matter has manifestations of waves as well as particles (in everyday life we don't usually notice the wave character of matter because the wavelengths are very long for macroscopic sized pieces of matter). Electrons are of course very small and their wave character can be extremely important -- allowed orbits are ones where an integral number of the waves associated with the electron can fit around the orbit (the electron is interfering constructively with itself):   

        electron.GIF (9514 bytes)

 

Spectral lines arise when electron jump from one allowed orbit to another:

    h_level.GIF (55174 bytes)

The energy of the emitted or absorbed photon equals the energy difference between the orbits. The lowest energy level orbit, lying closest to the nucleus, is called the ground state. Orbits are numbered starting with 1 for the ground state - the transition from the ground state to the next higher orbit would be denoted as 1 -> 2.

Can we compute what wavelengths would be seen in the spectrum from a hydrogen atom?

Yes but we will need to assume the following fact from quantum mechanics:

       

We need to compute the velocity of the electron:

       

The requirement that an integral number of waves fit around an orbits gives

       

or (after some algebra)

       

so these are the allowed radii for the electron orbits in a hydrogen atom with n=1 for the ground state, n=2 for the first excited state and so on

To compute the wavelengths of hydrogen's spectral lines, we need to compute the energies for the orbits with the radii specified above.

      

Using the values for v and r from above,

       

       

Note that

               

This is also denoted as R or Rydberg.

So the wavelengths will be

   

This relation gives the wavelengths that we can actually observe and hence is very useful!

Example:

Calculate the wavelength of light emitted when an electron goes from level n=2 to n=1:

       

This spectral line is also called Lyman-Image18_24.gif (868 bytes) and lies in the ultraviolet region.

Type of spectra

        spectra.gif (32865 bytes)

Continuous spectrum: the output of a blackbody; the output of a light bulb is of this type.

Emission-line spectrum: what is observed from a hot gas where electrons are predominately in excited states above the ground level. A "neon" light produces an emission-line spectrum.

Absorption spectrum: what is observed from cool gas lying between the observer and a continuum source. The cool gas has its electron predominately in the ground state.

        DISCRETB.GIF (4929 bytes)

Note that the wavelength of a spectral line is the same regardless of whether it is in emission or in absorption.

Recall another use of spectral lines:

If a source of electromagnetic radiation is moving, the wavelengths of emitted light will be shifted when observed:

    doppler.gif (4281 bytes)

If v is away from the observer, is positive ("redshifted")

and if v is towards the observe, is negative ("blueshifted").

This relation gives us a handy method for measuring the speeds of sources. In general, Doppler shifts are most easily measured using spectral lines.

pldoplrb.gif (5269 bytes)

The radial velocity (velocity along the line of sight) is computed using the observed wavelength for an emission line and the wavelength measured in the laboratory which is the wavelength for zero velocity.

For example, an cloud of hydrogen gas is emitting Lyman-Image18_24.gif (868 bytes) at a rest wavelength of 121.6 nm. The observed wavelength from this cloud is 122.3 nm. What is the radial velocity of this cloud?

Other Interactions Between Matter and Electromagnetic Radiation:

Photons can impart momentum to an object they hit (recall that momentum must be conserved).

Radiation pressure from photons (sunlight) can push small dust grains out of the Solar System:

Consider a small piece of dust at distance d from the Sun which is sphere of radius R and density :

To compute the force from radiation pressure, need to recall that a force can also be thought of as a change in momentum, p=mv. The momentum of a photon is

so the force on a particle is just the energy absorbed from photons divided by the speed of light.

The rate of energy absorption is just the flux of photons from the Sun that the particle absorbs:

Notice that this ratio is independent of distance from the Sun.

At some critical particle size Rcritical, the two forces will be equal and for all smaller particles, the radiation pressure will be larger causing such particles to be blown out of the Solar System:

If our dust particle is made of typical silicates with a density of 3000 kg/m3

so the existing particles of this size and smaller in the Solar System have been produced recently as the result of collisions between larger particles or disintegrations of larger objects.