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 As discussed
in Part 2,
Rutherford proposed
the Rutherford Model
of the Nuclear Atom,
with which the experimental data
of the alpha particle
scattering by atoms
was beautifully explained.
However, as mentioned
shortly on the page,
2-7: Summary of Part 2,
the Rutherford model
of the nuclear atom
was not able to explain
the stability of atoms
and the atomic spectra
standing on the viewpoint
of the classical theory.
Thus the classical theory
came to a deadlock
at this point.
[The Difficulty
in the Stability of Atoms]
It was considered
that,
in the Rutherford model
of the nuclear atom,
a heavy atomic nucleus
with the electric charge
+Ze, where Z
is the atomic number,
locates
at the center of an atom
and Z light electrons
are moving around the nucleus.
Let us explain
that this atomic structure
is quite unstable
in the classical theory.
According to Maxwellian
electromagnetism,
a charged particle
(an electron at present)
moving with an
acceleration
radiates electromagnetic
waves and
loses energy as
Here e is
the elementary charge
(the charge of an electron),
is the permeability
of free space,
and c is
the speed of light in vacuum.
The minus sign
on the right-hand side
implies
to lose the energy.
Let us think,
for simplicity,
of the hydrogen atom
in which
an electron
circulating about
a proton.
Since the mass
of a proton
is enough heavy
in comparison with that
of an electron,
the proton is assumed
to be at rest
at the origin of
the coordinate frame and
the electron is moving
along a circle of
the radius r
around the proton.
The Newtonian equation
of motion in the present case
is written
where the quantity
on the right-hand side
is the
strength of the Coulomb
force between
the proton and the electron.
The energy that
the electron loses
in unit time
is obtained from
Eqs. (1) and (2) as
On the other hand,
the energy of the electron
is written
We have therefore
The speed of the electron
would slow down,
because it loses energy.
Hence the radius
of its circular motion
decreases gradually.
Namely,
the radius r
is a decreasing
function of time.
Writing
and inserting
Eqs. (3) and (5) into
Eq. (6), we have
Let us suppose
that the radius
of a hydrogen atom
were
at a time.
It would be 0
after some time T
because of the loss
of energy of the electron.
We can get the time T
by integrating Eq. (7)
with respect to t
as
Thus,
the electron
in the Rutherford model
must rapidly
lose its energy
and spiral down
inward and collapse
on the proton
in an extremely
short time.
However, the real
hydrogen atom seems
quite stable
and does not collapse
spontaneously.
Thus, the Rutherford model
of the nuclear atom
brings about a
difficulty under
the classical theory.
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