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Author: G. Francesco Tartarelli
E-mail: [email protected]
Movie: 15 minutes in the life of the electron
Movie Clip From 14:36 To 17:36
Director: Luis Mariano Sesé Sánchez, José Antonio Tarazaga Blanco
Film Studio: UNED (Spain)
Advanced level
If we take a glass of water and we imagine to divide it in parts smaller and smaller
(which in practice becomes soon impossible) we can ask ourselves what’s the
smallest part we can get and that we can still call water. The answer is known: the
smallest part is what is called a molecule of water. If we divide a molecule in smaller
parts what we get is not water anymore. But can we really divide water in smaller
parts? The answer is yes. The water molecule is made of two atoms of hydrogen and
one atom of oxygen, as everybody knows: H2O. Neither hydrogen nor oxygen has
anything to do with water anymore; for example, in normal temperature and pressure
conditions they are both gases. If we want to continue our journey deeper in the
structure of matter we can still ask ourselves: can we divide an atom in smaller parts?
The answer is yes, again. Each atom is made by a central heavy part called nucleus
and by smaller particles, called electrons, that somehow move around the nucleus.
Can we subdivide the electrons and the nucleus further? Not the electrons. We finally
get to a true elementary particle: the electron is indivisible. Up to today there is no
indication at all that electrons could be subdivided further. This is not true for the
nucleus. The nucleus is a composite object made of particles called protons and
neutrons. The only exception is the hydrogen nucleus, which is composed by one
single proton. The story is already complicated but it complicates even further
because neither the protons nor the neutrons are elementary particles. They are made
by three smaller particles each, which are called quarks, Quarks, as electrons, are
true elementary particles.
But let’s stop here and take a look at the atom. How the nucleus and the electrons are
arranged inside the atom? A major advancement in these studies is due to the work
done by Ernest Rutherford and collaborators in the period going from 1906 to 1913. In
1906 Rutherford was studying the scattering of alpha particles (helium nuclei, made by
two protons and two neutrons) through thin metal (gold, silver) foils (about 4 m thick,
not enough to stop alpha particles), observing the spread of the beam on a
phosphorescent zinc sulfide screen. As expected a small scattering angle was
observed with most of the particles going through the foil directly (depending on the
thickness and material of the foil). However, in 1909, two Rutherford’s collaborators,
Hans Geiger and Ernest Marsden observed in such an experiment that in few
occasions alpha particles were scattered wih a scattering angle of more than 90 o
(that’s to say they were scattered backward). This was an unexpected and surprising
result. The scattering between the alpha particles and the target is a Coulomb
scattering: this means that it is due to the Coulomb forces between the charged alpha
particles and the charges inside the target atoms. At that time it was thought that the
atom followed Thomson’ s model which considered an atom as made of a sphere of
uniform positive charge in which the negative electrons were embedded (like raisins in
a cake). The small deflections of alpha particle trajectories usually observed were
explained like the result of many random collisions between the incident beam and the
atomic charges. This model, however, could not explain the rate of large scattering
angle observed, even summing several alpha particle-atom collisions. Rutherford
showed that the way to interpret the data would be to imagine the atom as made of a
small region of space (the nucleus) where basically all the atom mass is concentrated
carrying a positive charge surrounded by the electrons moving around the nucleus.
Large angle deflections would be explained by repulsion of alpha particles by the
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nucleus. This is more or less what happens in collisons between billiard balls. If you
throw a ball against another ball of the same mass which is at rest, there is no way the
incident ball will be scattered by more than 90o. You need a target ball with a much
higher mass then the incident ball to have such a large deflection. Using this model
Rutherford derived the mathematical expression of the cross section for such a
scattering and was able to successfully match the data obtained by Geiger and
Marsden.
Let’s consider the simplest atom, the hydrogen atom. The nucleus is made of a single
particle carrying a positive charge, the proton. If we consider the radius of the proton
charge we get a value of the order of 10-15 m = 1 fm. The mass of the proton is about
900 MeV. The hydrogen atom has a single electron which occupies a space having a
radius of about half an Angstrom (1 Angstrom = 10-10 m) around the nucleus. The
electron is an elementary particle: it has no internal structure, it is pointlike and it has a
mass of 0.5 MeV. It is clear from this figures that the mass of the hydrogen atom is the
mass of the nucleus while the dimension of the atom is determined by the radius at
which we can still find the electron charge. This means that the atom is basically
empty being the dimension of the nucleus a factor of 105 smaller than the size of the
atom. These last considerations also holds for heavier atoms, that’s to say atoms with
an higher number of protons (and electrons). However, the hydrogen atom is peculiar
in the sense that all other atoms have the nucleus made not only by protons but also
by neutrons. Neutrons carry no electrical charge and have a mass very close to that of
protons.
How are the electrons distributed around the nucleus? One of the most successfull
theories was the one proposed by Niels Bohr in 1915. It assumed that electrons are
rotating around the nucleus in circular orbits. For its similarity with the solar sistem it
was also called the „planetary model“ of the atom. However there are clear
differences. First of all the orbits in the atoms are not planar and also the force
keeping the electron on his orbit is not the gravitational force but the Coulomb
attractive force between the nuclues and the electron(s). But there is also a much
more striking difference. Let’s consider the hydrogen atom as an example as it is very
simple. What was discovered by Bohr studying the atomic spectra is that the atom
cannot take any energy value: it can only assume some discrete values. One says
that the energy is quantized. This implies that only some orbit radii are allowed. The
electron energies are given by the formula: E=-13.6 eV/n2 where n is an integer
number. The lowest energy level (n=1, radius closest to the nucleus) corresponds to
the hydrogen ground state and has an energy of E=-13.6 eV. Next level has an energy
of 3.4 eV (first excited level) and so on. As the quantum number n increases the
energy levels get closer together.
Despite its initial success, however, the Bohr model (and successive modifications)
has problem in explaining some observations including the spectra of atoms with
several electrons. The problem is due to the fact that particles like electrons or protons
are quantum objects and their behaviour inside an atom obeys the laws of quantum
mechanics. In this sense the Bohr model was a hybrid model: it incorporates some
new concepts like the quantization of energy but still uses classical concepts to
describe the motion of electrons inside the atom.
Based on the de Broglie relation p=h/, we have to assume that particles have also a
wave nature (and vice versa). If we can neglect these effects for everyday objects, we
cannot do this for electrons inside atoms. One of the main consequences is that we
have to abandon the idea of an electron as a localized particle described in classical
mechanics, for example, by a position vector and think it more as a wave.
This has several consequences; one of this is that we cannot specify the trajectory of
an electron like we do for a planet around a star or a billiard ball. We can only say
what is the probability to find an electron in a certain region of space. For example, for
a hydrogen atom there is spherical region around the nucleus where it is likely to find
an electron: more likely the electron will be close to the nucleus and the probability to
find it at a certain radius from the nucleus decreases with the distance from the
nucleus. These regions are called orbitals and replace the more common concept of
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orbits. All this might appear strange. However, the concepts of quantum mechanics
have been proved to be true by countless experiments. The fact is that our common
sense is driven by everyday experience based on the properties of macroscopic
objects. When we approach the atomic world and the world of elementary particles we
have to be ready to think in a different way.
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