Physics 321
Astrophysics II:  Lecture #5
Prof. Dale E. Gary

Classification of Spectra,
and Hertzsprung-Russell Diagrams

Spectra from the Hydrogen Atom

The Bohr model of the hydrogen atom gave us the following results: One way to see why the angular momentum should be quantized is to consider the electron as a wave.  As de Broglie showed, particles exhibit wave characteristics with wavelength given by
l = h/p = h/mv
What if the lowest orbit corresponded to one wavelength of such an electron wave?  The circumference of the orbit would correspond to a wavelength, so
2pr = l = h/mv  => l = mvr = h/2p.
The next higher orbit would correspond to two electron wavelengths, and the nth orbit to n electron wavelengths, so in this way we naturally find l = nh/2p.  If the orbit corresponded to a non-integer multiple of the electron wavelength, the electron wave would overlap and interfere with itself.

Here is a scale model of the orbits of the hydrogen atom, with the radii getting further apart according to n2:

Scale model of the hydrogen atom

However, despite the orbits getting ever farther apart in space, they get closer together in energy according to 1/n2.  Any energy level diagram is shown below:

Energy Level Diagram for Hydrogen
Excitation, De-excitation, Ionization and Recombination
When an electron jumps from a lower energy level to a higher energy level (smaller n to larger n), the atom is said to be excited, and the process is called excitation.  Energy must be added to the atom in order for such a jump to occur.  There are two ways for an atom to be excited: There are also two reverse processes by which the electron can jump from the higher energy level to a lower one, in the process of de-excitation.  These are: In addition to electrons jumping up and down between energy levels in the atom, it is possible for an electron to jump completely away from the atom (ionization), or a free electron to be captured into an energy level (recombination).
Heisenberg Uncertainty Principle
We noted last time the relationships:
DxDp ~ h/2p
DED t ~ h/2p
but did not have time to show some effects that come from them.  The n = 2 level of hydrogen lives for a very short time, only about 10-8 s.  That is, an electron in the first excited state will, after about 10-8 s, spontaneously emit a photon of energy 10.2 eV and go back to the ground state (emitting a Lyman a photon).  The second relation, above, means that the photons emitted have an energy uncertainty of
DE ~ h/2pD t = 1.05 x 10-26 J  =>  Dl ~ l2/2pD t c = 0.014 mA
This uncertainty in the wavelength shows up as a broadening of the spectral line (natural broadening).

Similarly, we can ask what is the uncertainty in speed that is required of an electron that is confined in a hydrogen atom (in the ground state orbit, with Bohr radius ao= 5.29 x 10-11 m).
Example: Imagine an electron comfined within a region of space the size of a hydrogen atom.  What is the minimum speed and energy of the electron, estimated using the Heisenberg uncertainty principle?  We know Dx ~ ao= 5.29 x 10-11 m so

Dp ~ h/2pao = 6.62 x 10-34 J s/2p(5.29 x 10-11 m) = 1.99 x 10-24 kg m/s.
This is the uncertainty in the momentum, which is about equal to the minimum momentum pmin that the electron has in order to be confined within the atom.  Now the momentum is just p = mv, so the minimum velocity is
vmin = pmin/me = 1.99 x 10-24 kg m/s / 9.1 x 10-31 kg = 2.19 x 106 m/s.
The corresponding minimum kinetic energy is
Kmin = 1/2mev2min= 2.18 x 10-18 J
which may seem like a small energy, but when converted to electron volts, this is
Kmin = 2.18 x 10-18 / 1.602 x 10-19 J/eV = 13.6 eV!
This is exactly the energy of an electron in the ground state of the hydrogen atom!  Later we will see that this subtle quantum effect is responsible for supporting the tremendous gravitational weight inside a white dwarf star.
Spectral Classification Strength of Spectral Lines Statistical Mechanics
These questions can be answered by appealing to statistical mechanics.  Imagine firing a beam of particles, all with the same energy, into a trap and allowing them to come into equilibrium by colliding elastically.  Due to standard probability and statistics (akin to coin-tossing), they will wind up with a gaussian distribution of speeds (a bell curve).  Any distribution of particles in equilibrium will have such a curve, which can be parametrized by a single number, the temperature.  Due to the 3D nature of velocity, the gaussian when expressed in terms of speed becomes:
nv dv = n (m/2pkT)3/2exp(-1/2 mv2/kT )(4pv2) dv
where the important dependence is the argument of the exponential, which is just the ratio of the kinetic energy (1/2 mv2) to thermal energy (kT ).  This expression is the number of particles per unit volume having speeds between v and v + dv.

Important relationships:

Boltzmann Equation
As atoms collide, their electrons can be knocked up to the next higher energy level if the colliding atoms have enough energy (collisional excitation).  The electron can even be knocked entirely away from the atom (ionization).  Again, looking at the problem from a statistical standpoint, the probability of the atom's being in one energy state, sa, is
P(sa) ~ exp(-Ea/kT )
and the probability for state sb is
P(sb) ~ exp(-Eb/kT )
where Ea and Eb are the energies of the two states (e.g.  = 13.6 eV for the ground state of the hydrogen atom).  The ratio of these probabilities is then
P(sb) exp(-Eb/kT )

P(sa) exp(-Ea/kT )

However, more than one state in an atom can have the same energy (i.e. they can be degenerate), e.g. in He I (neutral helium) two electrons in the ground state have n = 1, but the two electrons spin in opposite directions (ms = +1/2, -1/2).  We define the degeneracy (the number of states with the same energy Ea) as ga, which is called the statistical weight of state a.  The above expression then becomes

P(sb) gb exp(-Eb/kT )

P(sa) ga exp(-Ea/kT )

For a large number of atoms, the ratio of probabilities must be the same as the ratio of numbers of atoms in the two energy levels, e.g.

 N  g

 exp(-[Eb-Ea]/kT )
 N  ga

This is the Boltzmann Equation.  It turns out that for hydrogen, the statistical weights are given by gn= 2n2.  The Boltzmann Equation answers the first question we had, namely, how does the population of levels depend on T.  However, if we want to populate level 2 to make strong H-balmer lines, what temperature do we need, say, to make as many level 2 atoms as there are level 1 atoms?  The equation becomes

 N  8

 exp(-[E2-E1]/kT ) = 4 exp(-[-10.2 eV+13.6 eV]/kT ) = 1
 N  2

but when we solve for T we find T ~ 85,000 K!  Much higher than the observed 9,500 K.  Clearly there is another important effect that we did not include.  That other effect is ionization.

Saha Equation
To get a hydrogen atom's electron into level 2, it required a collision with at least 10.2 eV of energy.  However, once it is in level 2, it requires only 3.4 eV more energy to knock it completely away from the atom, so the first excited state is a very precarious place for an electron!  The actual number of atoms in level 2 at any one time is a balance between collisions to get it there in the first place, and collisions to ionize the atom.

Let ci be the ionization potential of the ith ionization stage.  This is the amount of energy needed to remove an electron from its lowest energy state.  For the ionization stage H I, for example, it would be 13.6 eV.  For He I it would be roughly the same, but for He II it would be four times larger (why?).  You might think that the ratio of atoms in one stage relative to another would be


 ~   exp(- ci / kT )

which is nearly right, but not all of the atoms are in this lowest state, so we have to add up statistically all of the ways an atom in various energy states can be stripped of an electron.

Partition Function
 Z  =   S g1 exp(-[Eb-Ea]/kT )

where the sum is over the distribution of energy states of an atom at temperature T from the Boltzmann Equation.



 exp(- ci / kT )

which leads to the Saha Equation

( 2pmekT )


 exp(- ci / kT )

Example: Saha Equation for H between 5,000 and 25,000 K.
Partition function for H II is ZII= 1, since there is no degeneracy for a bare proton.  For H I the summation can be done over just two terms (why?) which gives

ZII= 2 + 8 exp(-[(-10.2 eV)-(-13.6 eV)]/kT ) = 2 + 8 exp(-3.4 eV/kT )
Now, writing the thermal energy in eV, we have the range ~ 0.5 to 2.5 eV, so the exponential term ranges from about 8(0.001) to 8(0.25), so
ZII= 2 to 4
over the temperature range.  What we really want is not NII/NIas given by the Saha equation, but a related quantity NII/Ntotal = NII/(NI+NII) = NII/NI / (1+NII/NI) which is plotted in Figure 8-13 in the text (curve labelled log N+/No).  You can see that almost all of the H will be ionized by about 10,000 K.  This is what limits the strength of the H-balmer lines above 9,500 K.
Combining the Boltzmann and Saha Equations
We can find the variation of the strength of the H-alpha line with temperature by combining the Boltzmann and Saha equations as follows:




  NI+NII 1+N2/N1 1+NII/NI

The final expression is written entirely in terms of the quantities provided by the Boltzmann and Saha equations.  The full dependence is plotted in Figure below, and in Fig. 8-13 of the text.  Here we see very nice confirmation of the observed strength of the hydrogen lines, since the curve peaks right near 10,000 K as required to match the observations.

Ratio of hydrogen atoms in the n = 2 level to the total number of hydrogen atoms
as a function of frequency.  Figure from Carroll and Ostlie, Modern Astrophysics,
Addison-Wesley Publishing Co., 1996.