Arnold Sommerfeld assumed that the orbits of electrons doesn't have to be spherical but can also be elliptic. The electrons can move only on some, allowed ellipses. He coined a second l number which was called

**or the***the secondary quantum number***. The number defined the shape, the oblateness of an orbit. For n=1 the orbit can be only spherical (l=0), for n=2 there are two orbits of different shapes (l=0 - the elliptic one, l=1 - the spherical one). For any n there are n kinds of shapes of the orbits. The electrons moving on the two orbits of the same n number but of different shape have a bit different energies. That explaines the discovered structure of the spectral lines.***azimuthal quantum number* Another thing improving the Bohr atom was the discovery that the orbits don't have to lay in the same plane. They can be oriented in space on some defined directions. Their orientation is defined by

For a better understanding of the quantum numbers n,l,m

**m***the magnetic quantum number*_{l}. The electron circulating on an orbit causes the magnetic field. If the system is also placed in the outer magnetic field than the orbit of the electron places itself in a special way. That means that its position makes the direction and the sense of the magnetic field created by the electron the same as in the outer field. To deflect it to another position there should be some more energy given to the system. Sommerfeld proved that there is only some defined number of the possible orbit's positions. The number is equal 2*l+l. Each position in the magnetic field is of a bit different energy. The m number can name the values of 1 to -1. The phenomena of taking in the magnetic field the orbits of different energies for the same n is the explanation of the splitting of spectral lines - the Zeeman effect.For a better understanding of the quantum numbers n,l,m

_{l}the values for n=1, n=2, n=3, n=4 are given in the table.Beside the described facts the one more was discovered - that the spectral line consist of the two lines placed very close to each other - much closer than for the two shapes of orbits. The fact couldn't be explained by the Bohr-Sommerfeld model. Just the two Danish physicists - Uhlenbeck and Goudsmit explained the phenomena. The noticed that the electron circulates not only around the nucleus but also rotates. It can rotate in the two directions creating the rotary current flowing in the direction of the rotation. The current induces the magnetic field which is directed with the field created by the electron moving on the orbit or oppositely to it. So the fields sum or substract. The electron causing the oppositely directed field is of a bit smaller direction than the one causing the with directed field. So the spectral lines split. The spin, magnetic quantum number is used for defining the direction of rotation. The spin can be equal +1/2 or -1/2. That causes the noticed split of the spectral line on the two close placed ones.

After being improved the Bohr's theory very well described the spectrums of hydrogen. The main lines are caused by the electron taking different orbits (defined by the n). The lines consist of some very close placed lines for the different shapes of orbits (defined by the l). In the magnetic field the lines undergo a split because of the orbits taking defined planes in space. The last thing is that the electron can rotate in two directions what causes the split of the spectral lines on the two more ones.

The theory described the spectrum of hydrogen and the other hydrogen - like atoms (f. e. He

^{+}). It described the construction of atoms and of the orbits. So it was found a success. But for some scientists its assumptions looked artificial. The often asked question was: Why should the electrons circulate around the nucleus only on the defined orbits. The answer was given after 1925 by de Broglie, Schrodinger and Planck.

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