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Line Spectra

Atomic 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.

qd
qd
dqs

aa
[4] Hydrogen energy levels


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:

  1. The electron is able to move around the atomic nuclei on specific stable orbits without loosing its energy. With the orbital angular momentum aa
    r… orbit radius, m…electron mass, ω  … angular frequency,
  2. If the electron moves from an orbit with high energy to an orbit with lower energy, energy will be emitted.

Transition energy


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 · 108 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.

Hydrogen energy levels


 Z … charge of the nucleus (for hydrogen Z=1)
 ε0 … 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:     

Transition frequency

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.
qq

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):
qq
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. 

sd

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