electron orbits
atomic spectra
the Bohr atom
“How wonderful that we have met with a paradox. Now we have some
hope of making progress.”—Neils Bohr
So a “snapshot” of Rutherford’s atom
at some instant in time looks like this.
Of course, the electrons and nucleus are
very small, but I had to draw them big
enough to see.
-
++
++
-
Rutherford didn’t use the word “nucleus” in his 1911 paper
describing this model. He used the term “charge
concentration.”*
Rutherford was confident he had the atom figured out. "The
question of the stability of the atom proposed need not be
considered at this stage, for this will obviously depend upon
the minute structure of the atom, and on the motion of the
constituent charged parts."*
*See http://dbhs.wvusd.k12.ca.us/AtomicStructure/Rutherford-Model.html
4.2 Electron Orbits
Here’s our picture of Rutherford’s model
of the atom again. In the model,
electrons and nucleus are tiny and the
electrons are separated by large
distances from the positively charged
nuclei. Let’s build such an atom. Any
problems here?
-
++
++
-
The electrons would be sucked into the positively charged
nucleus (after all, unlike charges do attract, and the
electrostatic force is a very strong one).
Otherwise, they must be far enough away to not feel any
attraction, in which case they ought to wander off on their
own. Have you ever known an electron to just sit around and
do nothing?
-
Does anybody see another big
problem with this picture?
++
++
-
What holds the positive charges in the nucleus
together? After all, unlike charges repel.
This is a problem we won't solve until later in the course.
Forgetting about the nuclear problem for now, let’s play with
our atom builder a bit more and see if we can gain some
insight.
Dynamically stable orbits, like those of the
planets around the sun, would allow electrons
to remain "attached" to nuclei.
-
4+
-
That’s what Rutherford meant by “motion of the constituent
charged parts.” Let's calculate such an orbit.
As you learned in Physics 23, there must be a centripetal force
holding anything in circular motion (remember, circular motion
is accelerated motion): Fc = mv2/r.
In atoms, that force is the Coulomb attraction between
electrons and nuclei:
2
mv 2
e
FC =
=
.
2
4 ε 0r
r
What atom are
we talking about
here?
We can solve for electron velocity:
e
v=
.
4 ε0mr
Yes, hydrogen.
The electron's kinetic energy is K=mv2/2 and its potential
energy is
2
e
U=.
4 ε0r
Its total energy E is just E=K+U. Adding kinetic and potential
energies gives
2
e
E=.
8ε0r
The total energy is negative, meaning the electron is bound to
the nucleus. As Beiser points out, this total energy is actually
shared between electron and nucleus.
Check: use the ionization energy of hydrogen to calculate the
velocity and radius of the electron orbit. Do the results agree with
experiment?
The ionization energy is 13.6 eV, so the binding energy is EB =
-13.6 eV, or -2.2x10-18 Joules.
Solve the preceding equation for r:
r=-
e2
8 ε0EB
1.6×10 
== 5.3 × 10
 8  8.85×10  -2.2×10 
-19
-12
-18
-11
m
.
Using this r, we calculate the electron velocity to be 2.2x106
m/s. It is interesting that the electron velocity is far from
relativistic. This is not the case in heavier atoms.
Does anybody see problems with this orbiting-electron model?
This is a classical derivation, based on Newton's and Coulomb's
laws. It contradicts electromagnetic theory, which says that
the accelerated electron must radiate (i.e. lose) energy.
An accelerated electron
continuously radiates energy and
should spiral into the nucleus. Here
is another visualization.
Clearly, atoms do exist, so this
doesn’t happen.
The explanation? We simply can't use classical physics to
explain the atom. We are going to need to consider the wave
nature of electrons.*
*Of course, that was not an option in 1911 because de Broglie was just a
young undergraduate about to enter the army in World War I. De Broglie
postulated matter waves in 1924.
4.3 Atomic Spectra
In 1911, a young Danish physicist named
Neils Bohr had just earned his Ph.D. and
went to study with “Plum Pudding”
Thomsom. Unfortunately, Thomson was
not happy to hear that Bohr had ideas
that showed some of Thomson’s earlier
work to be wrong. Thomson didn’t have
much to do with Bohr.
In December 1911, Bohr met “Alpha Particle” Rutherford, who
he did get along with, and the next year went to work in his
lab.
The next few years saw rapid progress in understanding the
atom, and one key element was the study of atomic spectra.
The science of spectroscopy
probably began with Newton,
who showed that the spectrum of
light produced by a prism was
that of sunlight.
Here is a brief history of spectroscopy.
In the years around 1820,
Joseph Fraunhofer thoroughly
characterized the spectrum of
sunlight.
Between 1846 and 1849, bright line emission spectra from a
number of substances were identified and studied.
helium
mercury
hydrogen
increasing wavelength
increasing frequency
By 1871, Ångström had measured the wavelengths of the four
visible lines of the hydrogen spectrum.
In 1885, Balmer came up with an empirical formula (remember
what we think of those?) for the wavelengths of the hydrogen
lines.
1
1 1
= R  2 - 2  n = 3, 4,5,...
λ
2 n 
empirical parameters*
n=3
why start at 3?
n=4
n=5 n=6
What the heck does this mean? At the time, nobody had a
clue.
*fudge factors
Other scientists (Lyman, Paschen, Brackett, Pfund) came up
with similar formulas for infrared and ultraviolet hydrogen
spectral lines.
For your reading pleasure, these delectable formulas are reproduced in
your text on page 130.
1
1 1
=R 2 - 2 
λ
2 n 
n = 3,4,5,...
Balmer
1
1 1 
=R 2 - 2 
λ
1 n 
n = 2,3,4,...
Lyman
1
1 1
=R 2 - 2 
λ
3 n 
n = 4,5,6,...
Paschen
1
1 1
=R 2 - 2 
λ
4 n 
n = 5,6,7,...
Brackett
1
1 1
=R 2 - 2 
λ
5 n 
n = 6,7,8,...
Pfund
OK, so what’s the point?
There was a large collection of spectral data for hydrogen.
Only a single line could be explained by Thomson’s plum
pudding model. Rutherford’s “new and improved” model
predicted a continuous spectrum.
Wouldn’t it be a good idea to find a theory which
explains the data?
Absolutely! This section has been looking at the data. The
next section looks at the Bohr model which “explains” the
data.
One more thing… Beiser introduces the different kinds of
atomic spectra in this section.
http://csep10.phys.utk.edu/astr162/lect/light/absorption.html
Emission spectra: an atomic gas is excited, e.g., by an
electric current, and emits radiation of specific wavelengths.
Absorption spectra: an atomic gas absorbs radiation of
specific wavelengths.
Of course, molecules can have emission and absorption spectra too.
Summarizing section 4.3: if we put a high voltage current
through hydrogen gas, we see light.
background “glow” due to white
light leaking into spectrometer
The light is not a continuous spectrum, but a series of discrete,
monoenergetic lines.
Recall that the first series of lines was discovered by Balmer,
and the wavelengths are given by
1
1 1
=R 2 - 2 
λ
2 n 
n = 3, 4,5,...
R is a constant. Any good theory should be able to predict the
value of R, with no empirical parameters. This is an empirical
equation, which we would like to explain with the laws of
Physics. We can't explain it with classical physics.
I showed you other series of spectral lines. I am not
interested in having you memorize their names or formulas.
They all follow something like
1 1 
1
=R 2 - 2 
λ
 nf ni 
ni = nf +1, nf +2, nf +3, ...
The important idea is that we need to explain these
experimental observations, including the formula and the
spectra themselves. For those who enjoy sparkling lights, here
(dead link?) and here are some pages of spectra.
4.4 The Bohr Atom (1913)
The scenario that Beiser
presents in describing Bohr’s
model of hydrogen is not the
approach Bohr took in his
work, because de Broglie and
his matter waves didn't come
for another decade.
Nevertheless, let’s follow Beiser for a while, and at the end of
this section I’ll briefly mention Bohr’s approach.
“…if only sufficiently many physicists are placed at sufficiently big
machines, then everything will fall into place in the end.”—W. Heisenberg
(no, Heisenberg didn’t actually believe that)
Note that this is a model for the hydrogen atom.
Why choose the hydrogen atom...
Consider the wavelength of an orbiting electron.
An electron in orbit around a hydrogen nucleus has wavelength
 = h / mv and a velocity as given in section 4.2:
v=
e
.
4 0mr
You can solve these two equations for the wavelength:
h 4 ε0r
λ=
.
e
m
h 4 0r
λ=
.
e
m
You can plug in the electron mass, r = 5.3x10-11 m for the
radius of an electron orbit in hydrogen, e =1.6x10-19 C, and 0
= 8.85x10-12…
6.63 10-34 J  s
λ=
1.6 10-19 C
C2
-11
4  8.85 10

5.3

10
m
2
Nm
9.11 10-31 kg
-12
…and you get a wavelength  = 33x10-11 m, which
(coincidentally?) is the circumference of the electron orbit.
So what?
We have taken an electron and
calculated its wavelength. We
find that its wavelength exactly
corresponds to one orbit of the
hydrogen atom. See Figure 4.12.
Or try this visualization. (Click on the little “ball” to start.)
This gives us a clue as to how to construct electron orbits.
If an electron wave going around a
nucleus "meets itself" out of phase
after one revolution, destructive
interference will take place.
http://hyperphysics.phyastr.gsu.edu/hbase/ewav.html#c2
Actually, instead of "destroying" itself, such an electron simply
can't fit into an orbit. It must either gain or lose energy, so
as to have a velocity which gives it a wavelength which fits
into the orbit.
We thus arrive at the postulate that an electron can orbit a
nucleus only if its orbit contains an integral number of
de Broglie wavelengths.
Of course the idea of de Broglie waves was a decade in the
future when Bohr worked out his model for the hydrogen
atom, so he couldn't have made this postulate.
Expressed mathematically:
n λ = 2  rn
n = 1, 2, 3, ...
Combining this expression for  with the one we obtained
earlier in this section gives us an equation for rn, the orbital
radii in the Bohr atom.
n2h2ε0
rn =
 m2e
n = 1, 2, 3, ...
Like our particle in a box, an electron in hydrogen can “fit”
only if its wave can “fit.” The wavelength is quantized.
The orbital radii are quantized.
What else was quantized for our particle in box.
Energies!
I wonder if that will happen here…
(not a question, just
wondering out loud)
Here’s a visualization of the Bohr model.
If that visualization was supposed to mean something, there
are a couple of things we have to explain...
The integer n in the equations for  and r is called the
quantum number of the orbit.
When n=1,
2
r1 =
h ε0
 m2e
6.62×10   8.85×10 
=
= 5.292 × 10
  9.11×10 1.6×10 
-34 2
-31
-12
-19 2
-11
m
.
This value of r1 is the radius of the innermost orbit. It is called
the Bohr radius, a0, of the hydrogen atom, and a0=5.292x10-11
m. The other radii are given by rn = n2 a0.
If this wasn’t Bohr’s approach to his model of the atom, what
was?
As a student, Bohr hadn’t thought spectra were worth much.
However, when Bohr was trying to understand the hydrogen
atom, a colleague happened to show him some spectra and
Balmer’s formula.
1
1 1
= R  2 - 2  n = 3, 4,5,...
λ
2 n 
"As soon as I saw Balmer's formula, the whole thing
was immediately clear to me." –N. Bohr
Quoting from the class notes* of Taylor (U. of Virginia)…
*http://www.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html
“What he saw was that the set of allowed frequencies
(proportional to inverse wavelengths) emitted by the hydrogen
atom could all be expressed as differences.
This immediately suggested to him a generalization of his idea
of a "stationary state" lowest energy level, in which the
electron did not radiate.
There must be a whole sequence of these stationary states,
with radiation only taking place as the atom jumps from one to
another of lower energy, emitting a single quantum of
frequency f such that
h f = En – Em ,
the difference between the energies of the two states.
Did you spot the example of the misleading Physicsspeak* on
the previous slide?
Evidently, from the Balmer formula and its extension to general
integers m, n, these allowed non-radiating orbits, the
stationary states, could be labeled 1, 2, 3, ... , n, ... and had
energies -1, -1/4, -1/9, ..., -1/n2, ... in units of hcRH (using  f
= c and the Balmer equation above).”
The main thing is to realize that Bohr pictured a series of
“stationary states”* in which the electron could exist without
radiating, and that radiation takes place only when an electron
jumps between different states.
*How can an electron be orbiting if it is in a “stationary” state?
Furthermore, if, as we showed above, the speed is quantized,
then so is the electron’s angular momentum.
More specifically, it is not difficult to show that the angular
momentum is quantized in units of h/2, which we call ħ.
L = n h / 2 = n ħ .
I guess Beiser didn’t want to wrestle with angular momentum
in this chapter. I can’t say that I blame him.
Let’s think about Bohr’s model for a minute.
 It contains classical physics. Remember, we used our
“dynamically stable” orbit equations in the model.
e
v=
.
4 ε 0mr
 It doesn’t explain how an electron can exist in a “stationary
state” orbit without radiating. (Just giving it a name doesn’t
explain it.)
 It predicts Balmer’s (and all those others—see next section
in text) formula for the hydrogen spectral lines.
 But it doesn’t explain why the electron radiates when it
jumps between orbits.
Bohr’s model for hydrogen is a curious hybrid of classical and
quantum physics.
Some physics faculty refuse to teach it because it is wrong.
Several prominent physicists of Bohr’s day threatened to quit
the profession if Bohr was correct. “The prevailing impression
was one of scandal, or at least bewilderment, before the
undeserved success of such high-handed disregard of the
canons of formal logic.”*
*http://www.phys.virginia.edu/classes/252/Bohr_to_Waves/Bohr_to_Waves.html
"If you aren't confused by quantum physics, then you really haven't
understood it." –N. Bohr
Bohr himself admitted his model didn’t explain anything. But it
does tie together previously unexplained observations, and
tells us the direction we might go in looking for the “true”
model.
See here for another triumph* of Bohr’s model: singly ionized
helium (like hydrogen but with an extra neutron). However,
the model fails for helium. As we will see a few chapters from
now, that is nothing to be ashamed of.
There is a happy ending to this story: Bohr won the 1922
Nobel prize “for his services in the investigation of the
structure of atoms and of the radiation emanating from them.”
*"This is an enormous achievement. The theory of Bohr must then be
right." –A. Einstein