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:

N

**/ N**_{2 }**= [g**_{1 }**/ g**_{2}**]**_{1 }**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 N

**with energy E2, to the number o atoms N**_{2 }**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 g**_{1 }**degenerate states with energy E2 to the probability that the system is in any of the g**_{2 }**degenerate states E1, viz.**_{1 }
P(E2)

**/ P(E1)**_{ }**= [g**_{ }**/ g**_{2}**]**_{1 }**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
= g

**+ å**_{1}^{¥}_{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.
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

**= [1 - e**_{u }^{- h }^{u o}^{ / kT}] (**p**e^{2}/ mc) f f_{u}_{}

_{}

which
yields units in cm

^{2}/ 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

**multiplied by the number of absorbing atoms per unit volume. The value for k**_{u }_{u}is just multiplied by the number of absorbing atoms.
The value for a

**is just: a**_{o }**= a**_{o }**/**_{u }**f**_{ }_{u. }
Sometimes referred to as a “fudge factor”,
f is known as

**It is basically the transition probability for the line and is to be computed by quantum mechanics or measured in the laboratory.***the oscillator strength or f-value of the line.*
The broadening function f

_{u }is:
f

_{u }= 1/ Ö**p**[exp (u -u_{o})_{ }**/****D**u_{D}**]**^{2}D u_{D}_{}

Which
can also be rewritten as: f

_{u }du =
1/
Ö

**p**[exp (u - u_{o})_{ }/D u_{D}]^{2}du / D u_{D}_{}

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 u

_{D}is the**. As can be seen on inspection, f***Doppler half-width of the line*_{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.

*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 l

_{o}(undisturbed line) and the spread of wavelengths, D l:
D l = (l

_{o})^{2 }e H/ 4 π m_{e}c^{2 }
Where H is the intensity of the sunspot magnetic field in gauss (to be found), e is the electron charge in e.s.u., m

_{e}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: l

_{o }= 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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