Spectra from the Hydrogen Atom

The Bohr model of the hydrogen atom gave us the following results:

• The angular momentum, l, was postulated to be quantized according to
l = mvr = nh/2p. where n is the principle quantum number.
• The radii of allowed electron orbits were also quantized according
r = a_on² = n²h²/(4p²ke²m)
where a_o= 5.29 x 10^-11 m is the Bohr radius.
• The energies of the orbits were likewise quantized according to
E_n = -R' [1/n²]
where R' = 13.6 eV is a constant.  The electron-volt (eV) is a unit of energy (the energy an electron would gain after acceleration through a potential of 1 volt). 1 eV = 1.602 x 10^-19 J
• The difference between energy levels (orbits) corresponds to a wavelength
1/l_ab = n_ab/c = (E_b - E_a)/ch=R [1/n_b²-1/n_a²]
where R = 10.96776 mm^-1 is the Rydberg constant.

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:

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:

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:

• If the atom suffers a collision that causes an electron to jump from the lower to higher energy state it is called collisional excitation.
• If the electron captures a photon of energy hn corresponding to the energy difference between the two energy levels, and so jumps from the lower to higher energy state, it is called radiative excitation.

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:

• If the atom suffers a collision that causes an electron to jump from a higher to lower energy state it is called collisional de-excitation.
• If the atom jumps from a higher to lower energy state and emits a photon of energy hn corresponding to the energy difference between the two energy levels, it is called radiative de-excitation.  Since energy is released during such a transition, the transition is spontaneous, meaning it occurs without any outside action.

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).

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 a_o= 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 ~ a_o= 5.29 x 10^-11 m so

Dp ~ h/2pa_o = 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 p_min that the electron has in order to be confined within the atom.  Now the momentum is just p = mv, so the minimum velocity is

v_min = p_min/m_e = 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

K_min = 1/2m_ev²_min= 2.18 x 10^-18 J

which may seem like a small energy, but when converted to electron volts, this is

K_min = 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.

When spectra of stars are examined, spectral lines are easily seen, and the lines of the Balmer series for hydrogen are particularly recognizable.  The first spectral classification simply ordered stars according to how strong the H-Balmer lines were (Pickering and Fleming), as A, B, C, ...  See the spectra below.


Adapted from data in the electronic version of "A Library of Stellar Spectra,"
by Jacoby G.H., Hunter D.A., Christian C.A.  Astrophys. J. Suppl. Ser., 56, 257 (1984).

The spectra were then reordered according to temperature using the continuum spectrum (blackbody curve) as a guide (Annie Jump Cannon), to arrive at a new ordering:
 

O, B, A, F, G, K, M        (Oh Be A Fine Girl/Guy Kiss Me)
(early)         (late)

...B7, B8, B9, A0, A1, A2, A3, ..., A7, A8, A9, F0, F1... etc.
                       (early A)                  (late A)
 

It is puzzling, however, that the hydrogen lines should get stronger as we go into the sequence, then get weaker again.  They reach their greatest strength at A0, which is a blackbody temperature of about 9500 K.  Why should this be?

Background:
• The Balmer series are transitions from/to level 2 of the neutral hydrogen atom.
• We expect the lines to be strongest when lots of atoms have electrons in level 2 => high temperature.
• However, if the temperature is too high, the atom can be ionized (the electron is stripped off).
• Ionization notation:  H I     = neutral hydrogen
H II    = ionized hydrogen
He I   = neutral helium
He II  = singly-ionized helium
He III = doubly-ionized helium
These are called ionization stages.
To understand the behavior of the line strength with temperature, we must answer the following two questions:
• how does the population of levels depend on T?
• how does ionization depend on T?

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:

n_v 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:

• Peak of distribution (most probable speed) is when 1/2 mv² = kT, or
v_mp = [2kT/m]^1/2.
• Mean speed is somewhat higher (due to asymmetry in the curve)
v_mean = [3kT/m]^1/2.

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, s_a, is

P(s_a) ~ exp(-E_a/kT )

and the probability for state s_b is

P(s_b) ~ exp(-E_b/kT )

where E_a and E_b 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 (m_s = +1/2, -1/2).  We define the degeneracy (the number of states with the same energy E_a) as g_a, 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.
 

Nb		gb
	=		exp(-[Eb-Ea]/kT )
Na		ga

This is the Boltzmann Equation.  It turns out that for hydrogen, the statistical weights are given by g_n= 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
 

N2		8
	=		exp(-[E2-E1]/kT ) = 4 exp(-[-10.2 eV+13.6 eV]/kT ) = 1
N1		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.

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 c_i 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
 

Ni+1
	~	exp(- ci / kT )
Ni

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.

		infinity
Zi	=	S g1 exp(-[Eb-Ea]/kT )
		1=1

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

Then

Ni+1		Zi+j
	~		exp(- ci / kT )
Ni		Zi

which leads to the Saha Equation
 

Ni+1		2Zi+j	(	2pmekT	)
	=					exp(- ci / kT )
Ni		neZi		h2

Example: Saha Equation for H between 5,000 and 25,000 K.
Partition function for H II is Z_II= 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

Z_II= 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

Z_II= 2 to 4

over the temperature range.  What we really want is not N_II/N_Ias given by the Saha equation, but a related quantity N_II/N_total = N_II/(N_I+N_II) = N_II/N_I / (1+N_II/N_I) which is plotted in Figure 8-13 in the text (curve labelled log N₊/N_o).  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.

We can find the variation of the strength of the H-alpha line with temperature by combining the Boltzmann and Saha equations as follows:
 

N2		N2		NI
	=
Ntotal		N1+N2		Ntotal

or
 

N2		N2/N1		NI		N2/N1		1
	=				=
Ntotal		1+N2/N1		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.