Electron Orbits Binding Energy

Electron Orbits
• In an atom model in which negatively charged electrons move around a small positively
charged nucleus stable orbits are possible.
• Consider the simple example of an atom with a nucleus of charge of +e and one electron
with charge –e on an orbit around it (like in the hydrogen atom).
centrifugal force:
electrostatic force:
stability criterion:
Binding Energy
kinetic energy of the electron
on its orbit:
potential energy:
total energy:
experimental result for the binding
energy of Hydrogen (H):
binding energy
estimate of the radius of the electron
orbit:

Radiation of Electron on Orbit
electromagnetic power radiated by an
charge moving with acceleration :
centrifugal acceleration of electron on
circular orbit:
power radiated by electron in a hydrogen
atom (r ~ 0.05 nm):
• Under classical considerations this electron should loose its energy (-13,6 eV) very rapidly
and drop into the nucleus (which would be very bad news).
• Considering the wave-like properties of the electron however we will be able to explain (using
the laws of quantum mechanics) why the electron moves on a stable orbit around the
nucleus.
Emission and Absorption of Light
measuring the spectrum of light

Black Body Spectrum
• fusion (H 2 -> He)
• power ~ 100. 10 9 GW
• temperature
T ~ 6000 Kelvin
• continuous spectrum
• power on earth 1 kW/m 2
• largest intensity in visible part
of the spectrum
Spectrum of the Sun

Spectra
visible spectrum of hydrogen Na ()
Sodium Doublet
• transition between two electronic states with
different orbital angular momenta L
• evidence for electron spin due to spin-orbit
coupling visible in total angular momentum j
• naming convention of levels: n Lj

More Spectra
• the sun
• sodium (Na)
• mercury (Hg)
• lithium (Li)
• hydrogen (H)
Emission Spectra
• hot and dense objects (solids) display continuous spectra (e.g. the sun)
• the properties of a large number of interacting atoms (collisions) is observed
in this case
• frequently the spectrum can be explained using the ideas of black body
radiation (to be discussed later)
• the properties of individual atoms become apparent at low densities, when the
interactions are small
• then individual line spectra are observed
• the properties of these spectra are characteristic for the different elements
• both emission and absorption spectra can be observed

Spectral Series
• in the late 19 th century it was found that spectral
lines of simple elements can be ordered into simple
series
• one of the first ones was the Balmer (n=2) series of
spectral lines in Hydrogen in the visible wave
lengths.
n=5
n=4
n=3
n=2
Balmer series
n=1
Bohr Model
• If the electron orbit length would
not be an integer multiple
destructive interference would
appear and the orbit would not be
stable or even exist.

Quantum Numbers in the Bohr Model
Energy Levels of the Hydrogen Atom
total energy of electron as calculated before with n th Bohr
radius:
with Rydberg constant
The set of different energies are the of the
hydrogen atom. is the energy
corresponding to the quantum number . are
energies with the corresponding quantum
numbers.

Spectral Lines and Transitions in Hydrogen
Excitation, Relaxation and Lifetime
atoms are excited into higher energy states
• by collisions with electrons or other atoms(Franck-Hertz
experiment)
• by absorption of photons (c.f. absorption spectra)
after some time the atoms relax (decay) back to the ground state
the life time of an excited state
• is limited by spontaneous emission (coupling to vacuum
fluctuations, Einstein A coefficient to be discussed later)
• the emission can also be induced by interactions with other
photons, collisions with electrons or other atoms
• a wide range of life times is observed, a few nanoseconds to
minutes (if you are very careful)

Experiment of Franck and Hertz
vacuum tube with mercury (Hg) vapor
experimental set up
original measurement result
• The experiment demonstrates that atoms absorb energy from collisions with
electrons in quanta that are determined by the atoms energy level structure
given by the laws of quantum mechanics.
Nobel Prize in Physics (1925): Franck and Hertz
"for their discovery of the laws governing the
impact of an electron upon an atom"
James Franck
1/2 of the prize
Germany
Goettingen University
Goettingen, Germany
b. 1882
d. 1964
Gustav Ludwig Hertz
1/2 of the prize
Germany
Halle University
Halle, Germany
b. 1887
d. 1975

Experimental Setup
Spectrum of Neon
• excitation
by electron
collision
• highly excited electronic states
• transitions in the visible
frequency range
• lower excited electronic states
ground state

Comment of Franck on their Experiment
"It might interest you to know that when we made the experiments that we did not know Bohr's
theory. We had neither read nor heard about it. We had not read it because we were negligent to
read the literature well enough -- and you know how that happens. On the other hand, one would
think that other people would have told us about it. For instance, we had a colloquium at that
time in Berlin at which all the important papers were discussed. Nobody discussed Bohr's
theory. Why not? The reasons is that fifty years ago, one was so convinced that nobody would,
with the state of knowledge we had at that time, understand spectral line emission, so that if
somebody published a paper about it, one assumed, "Probably it is not right." So we did not
know it. But we made that experiment (and got the result that confirmed Bohr's theory) because
we hoped that if we found out where the borderline between elastic and inelastic impact lies ...
only one line might appear. But we did not know whether that would be so, and we did not know
whether at all an emission of an atom is of such a type that one line alone can be emitted and
all the energy can be used for that purpose. The experiment gave it to us, and we were surprised
about it. But we were not surprised after we read Bohr's paper later, after our publication."
-- Excerpt from one of three recordings of J. Franck, made in connection with a film on the Franck-Hertz
experiment at Educational Services, Inc., Watertown, Massachusetts, in January, 1961. As transcribed in
"On the recent past of physics", by Gerald Holton, American Journal of Physics, vol. 29, p. 805 (1961).
Correspondence Principle
• The prediction of quantum theories, like the (maybe too) simple Bohr model, should
correspond to the predictions of classical theories in the limit of large quantum numbers.
This fact is called the correspondence principle.

Hydrogen-Like Atoms
extension of Bohr model to other atoms with a single electron He + , Element (Z-1)+
energy levels in potential scaled with atomic number