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[1].
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.
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.
[1] 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.
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