In approaching stellar line formation we will be looking at a number of related equations, including:
1) The Boltzmann equation
2) The Saha equation
3) Combined Boltzmann and Saha equations
These will enable us to form a picture of spectral line formation which can then be generalized for different atoms and energy transitions. We start then with the Boltzmann equation, which we already introduced in the previous chapter:
N2 / N1 = [g2 / g1 ] exp (- E2 – E1) / kT
That is, for the atoms of a given element in a specified state of ionization, the ratio of the number of atoms N2 with energy E2, to the number of atoms N1 with energy E1, in different states of ionization is given by the above formula. The same form of the equation can also be used to find the ratio of probabilities, i.e. that the system will be found in any of the g2 degenerate states with energy E2 to the probability that the system is in any of the g1 degenerate states E1, viz.
P(E2) / P(E1) = [g2 / g1 ] exp (- E2 – E1) / kT
Thus, the Boltzmann equation can be posed in two forms. In statistical mechanics we could have also seen the partition function:
Z = å j exp ( - e j )/ t
Which is just the summation over the Boltzmann factor (exp ( - e j )/ t ) for all states j for which the number of particles (N) is constant. We will find it useful to rewrite it:
Z = g1 + å¥ j = 2 g j exp (- E j – E1) / kT
Of interest now are the relative numbers of atoms in ionization stage i, which is written:
N e N i + 1 / N i = 2 Z i + 1 / Z i (2 p m e kT/ h 2) 1.5 e - c i/ kT
This is the Saha equation, named after the Indian astrophysicist who first derived it. Here, N e is the number of free electrons per unit volume and c i is the ionization potential of the ith ionization stage. Thus, the equation relates the number of atoms in two successive ionization stages to the quantities that are relevant. As per our introduction to quantum mechanics, the factor ‘2’ in the equation refers to the two possible spins of the free hydrogen election with spin quantum number:
m s = +½.
m s = +½.
Recall that for thermodynamic equilibrium, the rate of ionization cannot exceed the rate of recombination. In other words, the rate at which atoms in the ith stage are ionized (i.e. to the i +1st stage) must equal the rate at wich ions in that i +1st stage are recombining with free electrons to form ions in the ith stage. The latter depends on N e N i + 1 and the former on N i . Hence, Saha’s equation simply expresses the fact these two processes must occur at the same rate.
log(N e N i + 1 /N i) =
15.38 + log (2 Z i + 1 / Z i ) + 1.5 log T – 5040 c i/ T
The units here are important to note, and are consistent with the ionization potential being measured in electron volts (eV). Therefore N e must be in particles per cubic centimeter.
Yet another way to express the Saha equation is to introduce the electron pressure, P e . This acknowledges that each separate species of particle makes its own contribution to the total gas pressure. The free electrons in a gas therefore produce a pressure given by: P e = N e kT.
Then we may write another log form of the Saha equation:
log(P e N i + 1 /N i) =
-0.48 + log (2 Z i + 1 / Z i ) + 2.5 log T – 5040 c i/ T
There are also two distinct processes by which lines can be formed:
1) Bound-bound transitions
2) Bound-free transitions
In the first case, the photon goes from one bound atom to another. Also, in case (1) the photon has a good chance of being scattered, i.e. emitted in the same downward transition. This may be at the same frequency as the absorption, in which case we say the scattering is coherent, or not, in which case it is non-coherent.
Details of bound-bound transitions differ in significant ways from bound-free transitions. In the first case, the transitions are also affected by a broadening function which is not so important for continuous emission. If we write out the equation for absorption in more detail we get:
a u = [1 - e - h u o / kT] (p e2/ mc) f f u
which yields units in cm2 / atoms at lower level. Two other absorption derivative values are possible from the preceding:
i) The absorption coefficient per unit length (cm-1)
ii) The mass absorption coefficient k u .
The value for (i) is just a u multiplied by the number of absorbing atoms per unit volume. The value for k u is just multiplied by the number of absorbing atoms.
The value for a o is just: a o = a u / f u.
Sometimes referred to as a “fudge factor”, f is known as the oscillator strength or f-value of the line. It is basically the transition probability for the line and is to be computed by quantum mechanics or measured in the laboratory.
The broadening function f u is:
f u = 1/ Öp [exp (u -u o) /D u D ]2 D uD
Which can also be rewritten as: f u du =
1/ Öp [exp (u - u o) /D uD ]2 du / D uD
This would be the probability that the absorbed photon lies between u and u + du, assuming equal intensities for all frequencies. Thus the integral:
ò f u du = 1
Where du is over all frequencies. The value of f u is larger near u o the frequency of the line center, as may be deduced from the line profile diagram below:
Absorption line profile showing the core and "wings"
Note here that D uD is the Doppler half-width of the line. As can be seen on inspection, f u is very large at line center and falls off in the “wings”, i.e. at larger and smaller frequencies.
The three important types of line broadening are doppler effect, natural and pressure broadening. We will confine our attention to the first type which is given by the broadening probability equation, provided the velocity is Maxwellian and that the frequency at line center u o is also observed for some u . The Maxwellian will display the distribution of velocities as shown below where the central line defines the most probable.
Maxwellian showing distribution of velocity with proportion of particles.
For the Sun, solar physics, it is also necessary to consider the Zeeman effect, a broadening due to strong magnetic fields such as in sunspots. An example of this applied to a sunspot is depicted below:
The left image shows the line-centered sunspot for which the Zeeman effect in classic "triplet" form (right image) is detected and measured. The greater the spectral line splitting the greater the magnitude of the associated magnetic field. George Ellery Hale, who discovered the effect, posed the quantitative relationship in terms of the original wavelength lo (undisturbed line) and the spread of wavelengths, D l:
D l = (lo)2 e H/ 4 π me c2
Where H is the intensity of the sunspot magnetic field in gauss (to be found), e is the electron charge in e.s.u., me is the mass of the electron in grams, and c the velocity of light in cm/ sec.
Problems:1) For a hydrogen plasma find:
(P e N i + 1 /N i) = P e N HII /N HI
at a temperature of 5040 K, given the hydrogen partition functions are:
Z i + 1 = Z 2 = 1 and:
Z i = Z 2 = 2 with c i = 13.6 eV
2) For the temperature and conditions of problem (1) of the previous set, find the ratio of the probability that the system will be found in any of the eight degenerate states of energy level E2, to the probability that the system will be in any of the two degenerate states of energy level E1.
3) An H-alpha line undergoes triplet splitting in the vicinity of a sunspot. The undisturbed line is measured at: lo = 6.62 x 10 -5 cm. The line shift on either side is: + 0.0025 Å. Use this information to find the strength of the sunspot magnetic field: a) in gauss, b) in Tesla.
 This must hold if the excitation and ionization equations are assumed valid, hence the numbers of atoms in a given level must not change with time.