Line SpectraAtomic Spectroscopy – How to describe a line spectrum
G. Kirchhoff, R.W. Bunsen (1859): Every element has its characteristic line spectrum.
If
you have a light source (e.g. mercury vapour lamp) and an instrument to
separate the light into different wavelengths (e.g. a grating
spectrograph), than you will see hopefully a line spectrum. The emitted
light comes from electron transitions between different energy levels
in the atom. At the beginning of the 20th century physicists
tried to understand this process. First they looked at an atom with
only one electron (hydrogen) and made empirical formulas to describe
the line series of the hydrogen line spectra (Lyman, Balmer, Paschen,
Brakett, Pfund).
Rydberg reformulated Balmer’s formula and found an equation which works also for the other series.
The “n” in those formulas is the so called “control variable” Later it turned out to be the principal quantum number.
1911
postulated Rutherford his atomic model in which he said the electron
moves around the nucleus (the positive charge being positioned in the
centre of the atom) on a circular path.
After this, Niels Bohr postulated in the year 1913:
 The
electron is able to move around the atomic nuclei on specific stable
orbits without loosing its energy. With the orbital angular momentum
r… orbit radius, m…electron mass, ω … angular frequency,
 If the electron moves from an orbit with high energy to an orbit with lower energy, energy will be emitted.
λ … wavelength
ν … frequency
n… principle quantum number
h … Planck’ s constant (= 6,626 · 10^{34} Js = 4,136 · 10^{15} eVs)
c … speed of light (= 2,998 · 10^{8} m/s)
This
model is called the “Bohr’s model”. With this model it is possible to
describe the energy of an electron in the coulomb field of the nucleus.
And so it is also possible to describe the energy difference
(wavelength) of a transition from a higher to a lower energy level
(orbit). The energy of an electron moving in the Coulomb field of the
nucleus of Z elemental charge units is given by.
Z … charge of the nucleus (for hydrogen Z=1)
ε_{0} … dielectric coefficient
If
the electron moves from orbit b (nb) to orbit a (na) with (nb>na), t
hen the frequency of the emitted light is given by:
Bohr’s
theory gives the right values for a one electron system, but without
fine structure. Fine structure means that the spectral lines are
consisting of several lines (depending on angular momentum and spin).
This was the classical approach to describe the energy levels in the
atom. Sommerfeld extended the theory of Bohr and he said that the
electron is moving on elliptical orbits around the nucleus. He needed
two quantum numbers to describe the energy levels (primary (n) and
secondary quantum number (l)).
But this wasn’t satisfying because:
 it
is a half classical theory  nearly without quantum mechanics, in
particular it does not solve a serious problem. In classical
electrodynamics, an electron moving on circles performs an accelerated
movement, and consequently must emit radiation and loose its energy;
 the atoms are considered as flat, tiny circles or ellipses;
 the theory works without the electron spin so that the fine structure is false;
 it doesn’t work for other atoms than hydrogen.
So the solution of all this gave Quantum mechanics (Schrödinger and
Heisenberg). The Schrödinger equation for an atom with one electron
gives the right energy values for the electron.
After solving the equation you will get for every wave function (depends on n and l) an Eigen value (energy level).
Experiments (Stern, Gerlach 1921) also have shown that the electron has
a spin (magnetic moment). The orbital angular momentum l of the
electron interacts with the electron spin s (ms = ± 1/2). So that the
solution of the problem is given by (with fine structure):
α is the so called fine structure constant that arises in the
Sommerfeld fine structure formula in which the motion [m = m(v)] of the
electron is relativistic treated.
Now it is obvious that the energy of an electron in a field of the nucleous depends on 3 quantum numbers (n, l, and s).
First published on the web: 11 March 2008
Authors : Lara Lobo Revilla and Karl Preiss . The text is based on a lecture given by Prof E.B. Steers at the first Gladnet training course in Antwerp Sept. 2007
