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43: The Bohr Model of Atoms 
The Rutherford model of the nuclear atom had been beautifully successful in explaining the alpha particle scattering. It was however completely powerless to explain the stability of atoms and to derive the atomic spectra. As seen in the previous sections, it brought about a serious difficulty which cannot be overcome within the classical theory. 
Discussing
with Rutherford in UK,
N. H. D. Bohr
(Denmark, 1885  1962)
learned the Rutherford model
and extended
this model to the socalled
Bohr model.
Bohr followed Rutherford and thought that, in a hydrogen atom, an electron revolves about the proton at such speed that the Coulomb force between the electron and proton becomes just equal to the centrifugal force to keep this orbital motion. As discussed on the preceding page, this model was unstable with respect to the classical theory. Moreover, this model had one more difficulty. It was known that all hydrogen atoms are equal to each other and have the equal radius, but there was nothing in this model to assure this equality. Thus, if the Rutherford model was to survive, it was necessary to find a radical remedy that would bring both the stability and the constant radius. 
To prepare the remedy,
taking a hint from
Balmer's formula
of the hydrogen spectra,
Bohr added the hypotheses
stated below to
the Rutherford model
(1913).
This model is
sometimes called
the BohrRutherford model
of the nuclear atom
or Bohr's quantum theory,
which was the first unified
theory to be able
to explain the
structure of the hydrogen atom
including both the stability
and the fixed radius.
[Bohr's Quantum Theory] Bohr considered that the electrons revolve around the atomic nucleus obeying the classical theory (Newtonian mechanics and Maxwellian electromagnetism). He added the following hypotheses to the Rutherford model: 
(1)  An atom can exist only in special states with discrete (steplike) values of energy. (Electrons in an atom can exist only in certain special orbits.) Bohr called these special states (or orbits) stationary states.  
(2) 
When an atom makes a
transition
from one stationary state
to another,
it emits or absorbs
radiation whose
frequency
_{},
is given by
the frequency condition
where E' and E'' are the energies of these stationary states. 

(3) 
In the stationary states,
the electrons move
obeying
the classical theory.
Among the possible motions
obeying the classical theory,
only those satisfying
the following
quantum condition
are realized:
where p is the momentum of the electron and q is its coordinate variable. The integration should be done along the orbit over one period of the cyclic motion. 
The theory constructed on the basis of the above hypotheses is called Bohr's quantum theory or sometimes the old quantum theory. 
[The Hydrogen Atom]
Let us apply Bohr's quantum theory to the hydrogen atom. Suppose that an electron (mass = m) revolves around the proton being at rest at the center under a Coulomb attractive force. This motion of the electron is described by the Newtonian equation of motion. The orbit of the electron in this case is elliptic in general. Now, let us assume it to be a circle for simplicity. In this case the absolute value of the momentum of the electron, p, is constant. Then the quantum condition is written where a is the radius of the orbit. From the balance of the centrifugal force and the Coulomb force, we have Combining Eqs. (3) and (4), we obtain the radius a of the orbit as Since the energy of the electron, E, is the sum of the kinetic and potential energies, we have Let us write this E_{n}. The allowed energies of the hydrogen atom are the discrete ones _{}, given by Eq. (6). The state of n = 1 is the lowest energy state called the ground state. The radius of the orbit in the ground state, a_{0}, is especially called the Bohr radius, whose value is This Bohr radius is thought to be the radius of an ordinary hydrogen atom, and there can exist no hydrogen atom with a radius smaller than this. When the hydrogen atom jumps (makes a transition) from one stationary state with the energy E_{n} to another with E_{k}, it emits or absorbs a light whose frequency _{}, is given by the frequency condition Eq. (1). Using the energy Eq. (6), we therefore have This is just the same as Balmer's Formula or Rydberg's formula which was obtained empirically. Therefore, the value of the Rydberg constant is calculated as This result is exactly fit to the experimental value. Thus, Bohr's quantum theory could beautifully reproduce the structure of the hydrogen atom. 
[The Energy Levels and
the Spectra of Hydrogen]
According to Bohr's quantum theory, the energy of the hydrogen atom is given by Eq. (6). The state of n = 1 is the ground state which has the lowest energy. The states of n = 2, 3, ... are the excited states. These energy levels are graphed in the following figure in units of eV, where the energy E = 0 corresponds to the ionized state of hydrogen with the free electron at rest. (The ionized state means the state in which the hydrogen nucleus, i.e. proton, and the electron are completely separated.) 
The Energy Levels of Hydrogen
The state of n = 1 is the ground state, and n = 2, 3, ... are the excited states. The energies are in units of eV, being so measured that the completely ionized state is made to be of 0 eV. 
The detailed experimental
data of the spectrum
of hydrogen are shown
in the following
figure,
where the Lymann, Balmer,
Paschen and Brackett series
are represented.
As seen in the
above figure,
these groups of spectral lines
are considered to
correspond to the
radiations emitted when
the transitions occur
from various states
to some specified state.

[The Confirmation
of the Stationary States]
In Bohr's quantum theory (or the Bohr model of atoms), Bohr postulated that an atom can exist only in the stationary states with discrete (steplike) values of energy. This postulation is quite difficult to be understood with the classical theory. However, the spectrum of the hydrogen atom could be reproduced beautifully. This means that the idea of the stationary states appears "correct". Therefore people wanted to confirm the existence of the stationary states directly by experiment. This is the famous FranckHertz experiment, which was carried out by J. Franck (Germany, USA, 1882  1964) and G. Hertz (Germany, 1887  1975). 
This Hertz is a researcher different from H. R. Hertz (Germany, 1857  94) who was famous for the research on electromagnetic waves in the 19th century. 
A schematic drawing
of the apparatus
used by Franck and Hertz
is shown in the following
Fig. (A).
Lowpressure mercury vapor
is introduced in a tube,
in which a filament F
is heated up
and electrons are
emitted from it.
The emitted electrons
are accelerated toward
the grid G
by the electric potential
between the filament F
and the grid G,
and after passing through
the grid have a
kinetic energy eV.
Then the electrons
are subjected to a small
retarding potential
_{}
before being collected
on the plate P.
If an electron loses no energy in the course from the filament to the grid, it will arrive at the plate with the energy _{} and contribute to the current measured by the ammeter A. However, if eV is greater than the excitation energy of the gas atom from the ground state to the first excited state, some of the electrons will lose a part of the kinetic energy on colliding with a gas atom. Then they have insufficient energy left to overcome the retarding potential _{} Consequently, as the potential V is increased gradually, we can find a quick drop of the current measured with the ammeter A, when the energy eV is equal to the excitation energy of the gas atom. If the potential V is then increased again, the current should rise, and the second drop would appear when eV is equal to twice the excitation energy. In 1914, Franck and Hertz demonstrated such energy absorption whose experimental data is shown in the above Fig. (B). As the potential V between the filament and the grid is raised, the current gradually increases, but suddenly decreases at V = 4.9V. After that, the current increases again and the second drop appears at V = 9.8V, and the similar phenomena are repeated. This means that the excitation energy of the mercury atom should be 4.9 eV. It was confirmed that this excitation energy was just equal to that obtained from the spectra of mercury. Franck and Hertz obtained similar experimental results using not only mercury but also neon, argon, krypton and so on. These results would be the experimental proofs of the Bohr's postulation of the stationary states. 
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