Dear Mark, Attached, please find final draft of the chapter on Nuclear Physics. Please, confirm receiving. Best wishes! Sergey Rakityansky
nuclear_final.ps
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\chapter{Inside atomic nucleus}
\label{ch_nucleus}
Amazingly enough, human mind that is kind of contained inside
a couple of liters of human's brain, is able to deal with extremely large
as well as extremely small objects such as the whole universe and its
smallest building blocks. So, what are these building blocks?
As we already know, the universe consists of galaxies, which consist of
stars with planets moving around. The planets are made of molecules,
which are bound groups (chemical compounds) of atoms.\\
There are more than $10^{20}$ stars in the universe. Currently,
scientists know over 12 million chemical compounds i.e. 12 million
different molecules. All this variety of molecules is made of only a
hundred of different atoms. For those who believe in beauty and
harmony of nature, this number is still too large. They would expect
to have just few different things from which all other substances are
made. In this chapter, we are going to find out what these elementary
things are.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{What the atom is made of}
\label{nucl_what_is}
The Greek word $\alpha\tau o\mu o\nu$ (atom) means indivisible. The
discovery of the fact that an atom is actually a complex system and
can be broken in pieces was the most important step and pivoting point
in the development of modern physics.\\
It was discovered (by Rutherford in 1911) that an atom consists of a
positively charged nucleus and negative electrons moving around it. At
first, people tried to visualize an atom as a microscopic analog of
our solar system where planets move around the sun. This naive
planetary model assumes that in the world of very small objects the
Newton laws of classical mechanics are valid. This, however, is not
the case.\\
%%%%%%%%%%%%%%%%%%%%%%%% FIGURE: Probability density in hydrogen %%%%%
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\caption
{\sf
Probability density $P(r)$ for finding the electron at a distance $r$
from the proton in the ground state of hydrogen atom.
}
\label{nucl_what_is_atom.fig}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The microscopic world is governed by quantum mechanics which does not
have such notion as trajectory. Instead, it describes the dynamics of
particles in terms of quantum states that are characterized by
probability distributions of various observable quantities.\\
For example, an electron in the atom is not moving along a certain
trajectory but rather along all imaginable trajectories with different
probabilities. If we were trying to catch this electron, after many
such attempts we would discover that the electron can be found anywere
around the nucleus, even very close to and very far from it. However,
the probabilities of finding the electron at different distances from
the nucleus would be different. What is amazing: the most probable
distance corresponds to the classical trajectory!\\
You can visualize the electron inside an atom as moving around the
nucleus chaotically and extremely fast so that for our ``mental eyes''
it forms a cloud. In some places this cloud is more dense while in other
places more thin. The density of the cloud corresponds to the probability
of finding the electron in a particular place. Space distribution of this
density (probability) is what we can calculate using quantum
mechanics. Results of such calculation for hydrogen atom are shown in
Fig. \ref{nucl_what_is_atom.fig}. As was mentioned above, the most
probable distance (maximum of the curve) coincides with the Bohr radius.\\
Quantum mechanical equation for any bound system (like an atom) can
have solutions only at a discrete set of energies $E_1, E_2, E_3
\dots$, etc. There are simply no solutions for the energies $E$ in
between these values, such as, for instance, $E_1<E<E_2$. This is why
a bound system of microscopic particles cannot have an arbitrary
energy and can only be in one of the quantum states. Each of such
states has certain energy and certain space configuration,
i.e. distribution of the probability.\\
A bound quantum system can make transitions from one quantum state to
another either spontaneously or as a result of interaction with other
systems. The energy conservation law is one of the most fundamental
and is valid in quantum world as well as in classical world. This
means that any transition between the states with energies $E_i$ and
$E_j$ is accompanied with either emission or absorption of the energy
$\Delta E=|E_i-E_j|$. This is how an atom emits light.\\
Electron is a very light particle. Its mass is negligible as compared
to the total mass of the atom. For example, in the lightest of all
atoms, hydrogen, the electron constitutes only 0.054\% of the atomic
mass. In the silicon atoms that are the main component of the rocks
around us, all 14 electrons make up only 0.027\% of the mass. Thus,
when holding a heavy rock in your hand, you actually feel the collective
weight of all the nuclei that are inside it.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nucleus}
\label{nucl_what_is.nucleus}
Is the nucleus a solid body? Is it an elementary building block of
nature? No and no! Although it is very small, a nucleus consists of
something even smaller.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proton}
\label{nucl_what_is.nucleus.proton}
The only way to do experiments with such small objects as atoms and
nuclei, is to collide them with each other and watch what happens.
Perhaps you think that this is a barbaric way, like colliding a
``Mercedes'' and ``Toyota'' in order to learn what is under their
bonnets. But with microscopic particles nothing else can be done.\\
In the early 1920's Rutherford and other physicists made many
experiments, changing one element into another by striking them with
energetic helium nuclei. They noticed that all the time hydrogen
nuclei were emitted in the process. It was apparent that the hydrogen
nucleus played a fundamental role in nuclear structure and was a
constituent part of all other nuclei. By the late 1920's physicists
were regularly referring to hydrogen nucleus as {\it proton}. The term
``proton'' seems to have been coined by Rutherford, and first
appears in print in 1920.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Neutron}
\label{nucl_what_is.nucleus.neutron}
Thus it was established that atomic nuclei consist of protons. Number
of protons in a nucleus is such that makes up its positive
charge. This number, therefore, coincides with the atomic number of
the element in the Mendeleev's Periodic table.\\
This sounded nice and logical, but serious questions remained. Indeed,
how can positively charged protons stay together in a nucleus?
Repelling each other by electric force, they should fly away in
different directions. Who keeps them together?\\
Furthermore, the proton mass is not enough to account for the nuclear
masses. For example, if the protons were the only particles in the
nucleus, then a helium nucleus (atomic number 2) would have two
protons and therefore only twice the mass of hydrogen. However, it
actually is four times heavier than hydrogen. This suggests that it
must be something else inside nuclei in addition to protons.\\
These additional particles that kind of ``glue'' the protons and make
up the nuclear mass, apparently, are electrically neutral. They were
therefore called {\it neutrons}. Rutherford predicted the existence
of the neutron in 1920. Twelve years later, in 1932, his assistant
James Chadwick found it and measured its mass, which turned out to be
almost the same but slightly larger than that of the proton.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Isotopes}
\label{nucl_what_is.nucleus.isotopes}
Thus, in the early 1930's it was finally proved that atomic nucleus
consists of two types of particles, the protons and neutrons. The
protons are positively charged while the neutrons are electrically
neutral. The proton charge is exactly equal but opposite to that of the
electron. The masses of proton and neutron are almost the same,
approximately 1836 and 1839 electron masses, respectively.\\
Apart from the electric charge, the proton and neutron have almost the
same properties. This is why there is a common name for them: {\it
nucleon}. Both the proton and neutron are nucleons, like a man and a
woman are both humans. In physics literature, the proton is denoted by
letter $p$ and the neutron by $n$. Sometimes, when the difference
between them is unimportant, it is used the letter $N$ meaning nucleon
(in the same sense as using the word ``person'' instead of man or
woman).\\
Chemical properties of an element are determined by the charge of its
atomic nucleus, i.e. by the number of protons. This number is called
the {\it atomic number} and is denoted by letter $Z$. The mass of an
atom depends on how many nucleons its nucleus contains. The number of
nucleons, i.e. total number of protons and neutrons, is called the
{\it atomic mass number} and is denoted by letter $A$.\\
Standard nuclear notation shows the chemical symbol, the mass number
and the atomic number of the isotope.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure isotope.fig %%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For example, the iron nucleus
(26-th place in the Mendeleev's periodic table of the elements) with
26 protons and 30 neutrons is denoted as
$$
^{56}_{26}{\rm Fe}\ ,
$$
where the total nuclear charge is $Z=26$ and the mass number $A=56$.
The number of neutrons is simply the difference $N=A-Z$ (here, it is
used the same letter $N$, as for nucleon, but this should not cause any
confusion). Chemical symbol is inseparably linked with $Z$. This is
why the lower index is sometimes omitted and you may encounter the
simplified notation like $^{56}{\rm Fe}$.\\
If we add or remove a few neutrons from a nucleus, the chemical
properties of the atom remain the same because its charge is the
same. This means that such atom should remain in the same place of the
Periodic table. In Greek, ``same place'' reads $\acute{\iota}\sigma
o\varsigma$ $\tau \acute{o}\pi o\varsigma$ (isos topos). The nuclei,
having the same number of protons, but different number of neutrons, are
called therefore {\it isotopes}.\\
Different isotopes of a given element have the same atomic number $Z$,
but different mass numbers $A$ since they have different numbers of
neutrons $N$. Chemical properties of different isotopes of an
element are identical, but they will often have great differences in
nuclear stability. For stable isotopes of the light elements, the number of
neutrons will be almost equal to the number of protons, but for heavier
elements, the number of neutrons is always greater than $Z$ and the
neutron excess tends to grow when $Z$ increases. This is because neutrons
are kind of glue that keeps repelling protons together. The greater the
repelling charge, the more glue you need.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nuclear force}
\label{nucl_forces}
Since atomic nuclei are very stable, the protons and neutrons must be
kept inside them by some force and this force must be rather
strong. What is this force? All of modern particle physics was
discovered in the effort to understand this force!\\
Trying to answer this question, at the beginning of the XX-th century,
physicists found that all they knew before, was inadequate. Actually,
by that time they knew only gravitational and electromagnetic forces.
It was clear that the forces holding nucleons were not
electromagnetic. Indeed, the protons, being positively charged, repel
each other and all nuclei would decay in a split of a second if some
other forces would not hold them together. On the other hand, it was
also clear that they were not gravitational, which would be too weak for
the task.\\
The simple conclusion was that nucleons are able to attract each other
by yet unknown {\it nuclear forces}, which are stronger than the
electromagnetic ones. Further studies proved that this hypothesis was
correct.\\
Nuclear force has rather unusual properties. Firstly, it is charge
independent. This means that in all pairs $nn$, $pp$, and $np$ nuclear
forces are the same. Secondly, at distances $\sim 10^{-13}$\,cm, the
nuclear force is attractive and very strong, $\sim100$ times stronger
than the electromagnetic repulsion. Thirdly, the nuclear force is of a
very short range. If the nucleons move away from each other for more
than few {\it fermi} (1\,fm=10$^{-13}$\,cm) the nuclear attraction
practically disappears. Therefore the nuclear force looks like a
``strong man with very short hands''.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Binding energy and nuclear masses}
\label{nucl_binding}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Binding energy}
\label{nucl_binding_energy}
When a system of particles is bound, you have to spend certain energy
to disintegrate it, i.e. to separate the particles. The easiest way to
do it is to strike the system with a moving particle that carries
kinetic energy, like we can destroy a glass bottle with a bullet or a
stone. If our bullet-particle moves too slow (i.e. does not have
enough kinetic energy) it cannot disintegrate the system. On the other
hand, if its kinetic energy is too high, the system is not only
disintegrated but the separated particles acquire some kinetic energy,
i.e. move away with some speed. There is an intermediate value of the
energy which is just enough to destroy the system without giving its
fragments any speed. This minimal energy needed to break up a bound
system is called {\it binding energy} of this system. It is usually
denoted by letter $B$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Nuclear energy units}
\label{nucl_binding_units}
The standart unit of energy, {\it Joule}, is too large to measure the
energies associated with individual nuclei. This is why in nuclear
physics it is more convenient to use a much smaller unit called
Mega-electron-Volt (MeV). This is the amount of energy that an
electron acquires after passing between two charged plates with the
potential difference (voltage) of one million Volts. Sounds very huge,
isn't it? But look at this relation
$$
1\,{\rm MeV}=1.602\times10^{-13}\,{\rm J}
$$
and think again. In the units of MeV, most of the energies in nuclear
world can be expressed by values with only few digits before decimal
point and without ten to the power of something. For example, the
binding energy of proton and neutron (which is the simplest nuclear
system and is called {\it deuteron}) is
$$
B_{pn}=2.225\,{\rm MeV}\ .
$$
The simplicity of the numbers is not the only advantage of using
the unit MeV. Another, more important advantage, comes from the fact
that most of experiments in nuclear physics are collision experiments,
where particles are accelerated by electric field and collide with
other particles. From the above value of $B_{pn}$, for instance, we
immediately know that in order to break up deuterons, we need to
bombard them with a flux of electrons accelerated through a voltage
not less than 2.225 million Volts. No calculation is needed! On the
other hand, if we know that a charged particle (with a unit charge)
passes through a voltage, say, 5 million Volts, we can, without any
calculation, say that it acqures the energy of 5\,MeV. It is very
convenient. Isn't it?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Mass defect}
\label{nucl_binding_defect}
Comparing the masses of atomic nuclei with the masses of the nucleons
that constitute them, we encounter a surprising fact: Total mass of
the nucleons is greater than mass of the nucleus! For example, for the
deuteron we have
$$
m_d<m_p+m_n\ ,
$$
where $m_d$, $m_p$, and $m_n$ are the masses of deuteron, proton, and
neutron, respectively. The difference is rather small,
$$
(m_p+m_n)-m_d= 3.968\times10^{-30}\,{\rm kg}\ ,
$$
but on the nuclear scale is noticeable since the mass of proton, for example,
$$
m_p=1672.623\times10^{-30}\,{\rm kg}
$$
is also very small. This phenomenon is called {\it ``mass defect''}.
Where the mass disappears to, when nucleons are bound?\\
To answer this question, we notice that the energy of a bound state is
lower than the energy of free particles. Indeed, to liberate them from
a bound complex, we have to give them some energy. Thinking in the
opposite direction, we conclude that, when forming a bound state, the
particles have to get rid of the energy excess, which is exactly equal
to the binding energy. This is observed experimentally: When a proton
captures a neutron to form a deuteron, the excess energy of 2.225\,MeV
is emitted via electromagnetic radiation.\\
A logical conclusion from the above comes by itself: When proton and
neutron are bounding, some part of their mass disappears together with
the energy that is carried away by the radiation. And in the opposite
process, when we break up the deutron, we give it the energy, some
part of which makes up the lost mass.\\
Albert Einstein came to the idea of the equivalence between the mass
and energy long before any experimental evidences were found. In his
theory of relativity, he showed that total energy $E$ of a moving body
with mass $m$ is
\begin{equation}
\label{nucl.emc2tot}
E=\displaystyle\frac{mc^2}{\sqrt{1-\displaystyle\frac{v^2}{c^2}}}\ ,
\end{equation}
where $v$ is its velocity and $c$ the speed of light. Applying this
equation to a non-moving body ($v=0$), we conclude that it possesses
the {\it rest} energy
\begin{equation}
\label{nucl.emc2}
E_0=mc^2
\end{equation}
simply because it has mass. As you will see, this very formula is the
basis for making nuclear bombs and nuclear power stations!\\
All the development of physics and chemistry, preceding the theory of
relativity, was based on the assumption that the mass and energy of a
closed system are conserving in all possible processes and they are
conserved separately. In reality, it turned out that the conserving
quantity is the {\it mass-energy},
$$
E_{\rm kin}+E_{\rm pot}+E_{\rm rad}+mc^2={\rm const}\ ,
$$
i.e. the sum of kinetic energy, potential energy, the energy of
radiation, and the mass of the system.
In chemical reactions the fraction of the mass that is transformed
into other forms of energy (and vise versa), is so small that it is
not detectable even in most precise measurements. In nuclear
processes, however, the energy release is very often millions times
higher and therefore is observable.\\
You should not think that mutual transformations of mass and energy
are the features of only nuclear and atomic processes. If you break up
a piece of rubber or chewing gum, for example, in two parts, then the
sum of masses of these parts will be slightly larger than the mass of
the whole piece. Of course we will not be able to detect this ``mass
defect'' with our scales. But we can calculate it, using the Einstein
formula (\ref{nucl.emc2}). For this, we would need to measure somehow
the mechanical work $A$ used to break up the whole piece (i.e. the
amount of energy supplied to it). This can be done by measuring the
force and displacement in the breaking process. Then, according to
Eq. (\ref{nucl.emc2}), the mass defect is
$$
\Delta m=\frac{A}{c^2}\ .
$$
To estimate possible effect, let us assume that we need to stretch a
piece of rubber in 10\,cm before it breaks, and the average force
needed for this is 10\,N (approximately 1 kg). Then
$$
A=10\,{\rm N}\times 0.1\,{\rm m}=1\,{\rm J}\ ,
$$
and hence
$$
\Delta m=\frac{1\,{\rm J}}{(299792458\,{\rm m/s})^2}\approx
1.1\times10^{-17}\,{\rm kg}.
$$
This is very small value for measuring with a scale, but huge as
compared to typical masses of atoms and nuclei.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Nuclear masses}
\label{nucl_binding_masses}
Apparently, an individual nucleus cannot be put on a scale to measure
its mass. Then how can nuclear masses be measured?\\
This is done with the help of the devices called {\it mass spectrometers}.
In them, a flux of identical nuclei, accelerated to a certain energy,
is directed to a screen where it makes a visible mark. Before striking
the screen, this flux passes through magnetic field, which is
perpendicular to velocity of the nuclei. As a result, the flux is
deflected to certain angle. The greater the mass, the smaller is the
angle (because of inertia). Thus, measuring the displacement of the
mark from the center of the screen, we can find the deflection angle
and then calculate the mass.\\
Since mass and energy are equivalent, in nuclear physics it is
customary to measure masses of all particles in the units of energy,
namely, in MeV. Examples of masses of subatomic particles are given in
Table \ref{tabl.nucl.nuclear_masses}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[!h!t!b]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
particle & number of protons & number of neutrons & mass (MeV)\\
\hline
$e$ & 0 & 0 & 0.511\\
\hline
$p$ & 1 & 0 & 938.272\\
\hline
$n$ & 0 & 1 & 939.566\\
\hline
$^2_1$H & 1 & 1 & 1875.613\\
\hline
$^3_1$H & 1 & 2 & 2808.920\\
\hline
$^3_2$He & 2 & 1 & 2808.391\\
\hline
$^4_2$He & 2 & 2 & 3727.378\\
\hline
$^7_3$Li & 3 & 4 & 6533.832\\
\hline
$^9_4$Be & 4 & 5 & 8392.748\\
\hline
$^{12}_{\phantom{0}6}$C & 6 & 6 & 11174.860\\
\hline
$^{16}_{\phantom{0}8}$O & 8 & 6 & 14895.077\\
\hline
$^{238}_{\phantom{0}92}$U & 92 & 146 & 221695.831\\
\hline
\end{tabular}
\caption{\sf Masses of electron, nucleons, and some nuclei.}
\label{tabl.nucl.nuclear_masses}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The values given in this table, are the energies to which the nuclear
masses are equivalent via the Einstein formula (\ref{nucl.emc2}).\\
There are several advantages of using the units of MeV to measure
particle masses. First of all, like with nuclear energies, we avoid
handling very small numbers that involve ten to the power of
something. For example, if we were measuring masses in kg, the
electron mass would be $m_e=9.1093897\times10^{-31}$\,kg. When masses
are given in the equivalent energy units, it is very easy to calculate
the mass defect. Indeed, adding the masses of proton and neutron, given
in the second and third rows of Table \ref{tabl.nucl.nuclear_masses},
and subtracting the mass of $^2_1$H, we obtain the binding energy
2.225\,MeV of the deuteron without further ado. One more advantage
comes from particle physics. In collisions of very fast moving
particles new particles (like electrons) can be created from vacuum,
i.e. kinetic energy is directly transformed into mass. If the mass is
expressed in the energy units, we know how much energy is needed to
create this or that particle, without calculations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Radioactivity}
\label{nucl_decays}
As was said before, the nucleus experiences the intense struggle
between the electric repulsion of protons and nuclear attraction of
the nucleons to each other. It therefore should not be surprising that
there are many nuclei that are unstable. They can spontaneously
(i.e. without an external push) break in pieces. When the fragments
reach the distances where the short range nuclear attraction
disappears, they fiercely push each other away by the electric
forces. Thus accelerated, they move in different directions like small
bullets making destruction on their way. This is an example of nuclear
radioactivity but there are several other varieties of radioactive
decay.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discovery of radioactivity}
\label{nucl_decays_discovery}
Nuclear radioactivity was discovered by Antoine Henri Becquerel in
1896. Following Wilhelm Roentgen who discovered the X-rays, Becquerel
pursued his own investigations of these mysterious rays.\\
The material Becquerel chose to work with contained uranium. He found
that the crystals containing uranium and exposed to sunlight, made
images on photographic plates even wrapped in black paper. He
mistakingly concluded that the sun's energy was being absorbed by the
uranium which then emitted X-rays. The truth was revealed thanks to
bad weather.\\
On the 26th and 27th of February 1896 the skies over Paris were
overcast and the uranium crystals Becquerel intended to expose to the
sun were returned to a drawer and put over (by chance) the
photographic plates. On the first of March, Becquerel developed the
plates and to his surprise, found that the images on them were clear
and strong. Therefore the uranium emitted radiation without an
external source of energy such as the sun. This was the first
observation of the nuclear radioactivity.\\
Later, Becquerel demonstrated that the uranium radiation was similar
to the X-rays but, unlike them, could be deflected by a magnetic field
and therefore must consist of charged particles. For his discovery of
radioactivity, Becquerel was awarded the 1903 Nobel Prize for physics.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Nuclear $\alpha$, $\beta$, and $\gamma$ rays}
\label{nucl_decays_alpha_beta_gamma}
Classical experiment that revealed complex content of the nuclear
radiation, was done as follows. The radium crystals (another
radioactive element) were put at the bottom of a narrow straight
channel made in a thick piece of lead and open at one side. The lead
absorbed everything except the particles moving along the channel.
This device therefore produced a flux of particles moving in one
direction like bullets from a machine gun. In front of the channel
was a photoplate that could register the particles.\\
Without the magnetic field, the image on the plate was in the form of
one single dot. When the device was immersed into a perpendicular
magnetic field, the flux of particles was split in three fluxes, which
was reflected by three dots on the photographic plate.\\
One of the three fluxes was stright, while two others were deflected in
opposite directions. This showed that the initial flux contained positive,
negative, and neutral particles. They were named respectively the $\alpha$,
$\beta$, and $\gamma$ particles.\\
The $\alpha$-rays were found to be the $^4$He nuclei, two protons and
two neutrons bound together. They have weak penetrating ability, a few
centimeters of air or a few sheets of paper can effectively block
them. The $\beta$-rays proved to be electrons. They have a greater
penetrating power than the $\alpha$-particles and can penetrate 3\,mm
of aluminum. The $\gamma$-rays are not deflected because they are high
energy photons. They have the same nature as the radio waves, visible
light, and the X-rays, but have much shorter wavelength and therefore
are much more energetic. Among the three, the $\gamma$-rays have the
greatest penetrating power being able to pass through several
centimeters of lead and still be detected on the other side.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Danger of the ionizing radiation}
\label{nucl_decays_danger}
The $\alpha$, $\beta$, and $\gamma$ particles moving through matter,
collide with atoms and knock out electrons from them, i.e. make
positive ions out of the atoms. This is why these rays are called {\it
ionizing radiation}.\\
Apart from ionizing the atoms, this radiation destroys molecules. For
humans and all other organisms, this is the most dangerous feature of
the radiation. Imagine thousands of tiny tiny bullets passing through
your body and making destruction on their way. Although people do not
feel any pain when exposed to nuclear radiation, it harms the cells of
the body and thus can make people sick or even kill them. Illness can
strike people years after their exposure to nuclear radiation. For
example, the ionizing particles can randomly modify the DNA (long
organic molecules that store all the information on how a particular
cell should function in the body). As a result, some cells with wrong
DNA may become cancer cells.\\
Fortunately, our body is able to repair some damages caused by
radiation. Indeed, we are constantly bombarded by the radiation
coming from the outer space as well as from the inner parts of our own
planet and still survive. However, if the number of damages becomes
too large, the body will not cope with them anymore.\\
There are established norms and acceptable limits for the radiation
that are considered safe for human body. If you are going to work in
contact with radioactive materials or near them, make sure that the
exposure dose is monitored and the limits are adhered to.\\
You should understand that no costume can protect you from
$\gamma$-rays! Only a thick wall of concrete or metal can stop them.
The special costumes and masks that people wear when handling
radioactive materials, protect them not from the rays but from
contamination with that materials. Imagine if few specks of
radioactive dirt stain your everyday clothes or if you inhale
radioactive atoms. They will remain with you all the time and will
shoot the ``bullets'' at you even when you are sleeping.\\
In many cases, a very effective way of protecting yourself from the
radiation is to keep certain distance. Radiation from nuclear sources
is distributed equally in all directions. Therefore the number $n$ of
dangerous particles passing every second through a unit area (say 1
cm$^2$) is the total number $N$ of particles emitted during 1 second,
divided by the surface of a sphere
$$
n=\frac{N}{4\pi r^2}\ ,
$$
where $r$ is the distance at which we make the observation. From this
simple formula, it is seen that the radiation intensity falls down
with incresing distance quadratically. In other words, if you increase
the distance by a factor of 2, your exposure to the radiation will be
decreased by a factor of 4.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Decay law}
\label{nucl_decays_law}
Unstable nuclei decay spontaneously. A given nucleus can decay next
moment, next day or even next century. Nobody can predict when it is
going to happen. Despite this seemingly chaotic and ``unscientific''
situation, there is a strict order in all this.\\
Atomic nuclei, being microscopic objects, are ruled by quantum
probabilistic laws. Although we cannot predict the exact moment of its
decay, we can calculate the probability that a nucleus will decay
within this or that time interval. Nuclei decay because of their
internal dynamics and not because they become ``old'' or somehow
``rotten''.\\
To illustrate this, let us imagine that yesterday morning we found
that a certain nucleus was going to decay within 24 hours with the
probability of 50\%. However, this morning we found that it is still
``alive''. This fact does not mean that the decay probability for
another 24 hours increased. Not at all! It remains the same, 50\%,
because the nucleus remains the same, nothing wrong happened to
it. This can go on and on for centuries.\\
Actually, we never deal with individual nuclei but rather with huge
numbers of identical nuclei. For such collections (ensembles) of
quantum objects, the probabilistic laws become statictical laws. Let
us assume that in the above example we had 1 million identical nuclei
instead of only one. Then by this morning only half of these nuclei
would survive because the decay probability for 24 hours was
50\%. Among the remaining 500000 nuclei, 250000 will decay by tomorrow
morning, then after another 24 hours only 125000 will remain and so
on.\\
The number of unstable nuclei that are still ``alive'' continuously
decreases with time according to the curve shown in
Fig. \ref{nucl_decays_law.fig1}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure nucl_decays_law.fig1 %%%%%%%%%%%%%%%
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\caption{\sf
The time $T_{1/2}$ during which one half of the initial amount of
unstable particles decay, is called their half-life time.
}
\label{nucl_decays_law.fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If initially, at time $t=0$, their number is $N_0$, then after certain
time interval $T_{1/2}$ only half of these nuclei will remain, namely,
$\frac12 N_0$. Another one half of the remaining half will decay
during another such interval. So, after the time $2T_{1/2}$, we will
have only one quarter of the initial amount, and so on. The time
interval $T_{1/2}$, during which one half of unstable nuclei decay, is
called their {\it half-life time}. It is specific for each unstable
nucleus and vary from a fraction of a second to thousands and millions
of years. A few examples of such lifetimes are given in Table
\ref{tabl.nucl.lifetimes}
\begin{table}[!h!t!b]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{\sf isotope} & $T_{1/2}$ & {\sf decay mode}\\
\hline
$^{214}_{\phantom{0}84}$Po & $1.64\times10^{-4}$\,s & $\alpha, \gamma$\\
\hline
$^{89}_{36}$Kr & $3.16$\,min & $\beta^-, \gamma$\\
\hline
$^{222}_{\phantom{0}86}$Rn & 3.83\,days & $\alpha, \gamma$\\
\hline
$^{90}_{38}$Sr & $28.5$\,years & $\beta^-$\\
\hline
$^{226}_{\phantom{0}88}$Ra & $1.6\times10^{3}$\,years & $\alpha, \gamma$\\
\hline
$^{14}_{\phantom{0}6}$C & $5.73\times10^{3}$\,years & $\beta^-$\\
\hline
$^{238}_{\phantom{0}92}$U & $4.47\times10^{9}$\,years & $\alpha, \gamma$\\
\hline
$^{115}_{\phantom{0}49}$In & $4.41\times10^{14}$\,years & $\beta^-$\\
\hline
\end{tabular}
\caption{\sf Half-life times of several unstable isotopes.}
\label{tabl.nucl.lifetimes}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Radioactive dating}
\label{nucl_decays_dating}
Examining the amounts of the decay products makes possible radioactive
dating. The most famous is the {\it Carbon dating}, a variety of
radioactive dating which is applicable only to matter which was once
living and presumed to be in equilibrium with the atmosphere, taking
in carbon dioxide from the air for photosynthesis.\\
Cosmic ray protons blast nuclei in the upper atmosphere, producing
neutrons which in turn bombard nitrogen, the major constituent of the
atmosphere. This neutron bombardment produces the radioactive isotope
$^{14}_{\phantom{0}6}$C. The radioactive carbon-14 combines with
oxygen to form carbon dioxide and is incorporated into the cycle of
living things.\\
The isotope $^{14}_{\phantom{0}6}$C decays (see Table
\ref{tabl.nucl.lifetimes}) inside living bodies but is replenished
from the air and food. Therefore, while an organism is alive, the
concentration of this isotope in the body remains constant. After death,
the replenishment from the breath and food stops, but the isotopes that
are in the dead body continue to decay. As a result the concentration of
$^{14}_{\phantom{0}6}$C in it gradually decreases according to the
curve shown in Fig. \ref{nucl_decays_law.fig1}. The time $t=0$ on this
Figure corresponds to the moment of death, and $N_0$ is the equilibrium
concentration of $^{14}_{\phantom{0}6}$C in living organisms.\\
Therefore, by measuring the radioactive emissions from once-living
matter and comparing its activity with the equilibrium level of
emissions from things living today, an estimation of the time elapsed
can be made. For example, if the rate of the radioactive emissions
from a piece of wood, caused by the decay of $^{14}_{\phantom{0}6}$C,
is one-half lower than from living trees, then we can conclude that we
are at the point $t=T_{1/2}$ on the curve \ref{nucl_decays_law.fig1},
i.e. it is elapsed exactly one half-life-time period. According to the
Table \ref{tabl.nucl.lifetimes}), this means that the tree, from which
this piece of wood was made, was cut approximately 5730 years ago.
This is how physicists help archaeologists to assign dates to various
organic materials.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nuclear reactions}
\label{nucl_reactions}
Those of you who studied chemistry, are familiar with the notion of
chemical reaction, which, in essence, is just regrouping of atoms that
constitute molecules. As a result, reagent chemical compounds are
transformed into product compounds.\\
In the world of nuclear particles, similar processes are
possible. When nuclei are close to each other, nucleons from one
nucleus can ``jump'' into another one. This happens because there are
attractive and repulsive forces between the nucleons. The complicated
interplay of these forces may cause their regrouping. As a result,
the reagent particles are transformed into product particles. Such
processes are called {\it nuclear reactions}.\\
For example, when two isotopes $^3_2$He collide, the six nucleons
constituting them, can rearrange in such a way that the isotope
$^4_2$He is formed and two protons are liberated. Similarly to
chemical reactions, this process is denoted as
\begin{equation}
\label{nucl_reactions.3he3he}
{}^3_2{\rm He} + {}^3_2{\rm He} \longrightarrow
{}^4_2{\rm He} + p + p + 12.86\,{\rm MeV}\ .
\end{equation}
The same as in chemical reactions, nuclear reactions can also be
either exothermic (i.e. releasing energy) or endothermic
(i.e. requiring an energy input). The above reaction releases
12.86\,MeV of energy. This is because the total mass on the left hand
side of Eq. (\ref{nucl_reactions.3he3he}) is in 12.86\,MeV greater
than the total mass of the products on the right hand side (you can
check this using Table \ref{tabl.nucl.nuclear_masses}).\\
Thus, when considering a particular nuclear reaction, we can always
learn if it releases or absorbs energy. For this, we only need to
compare total masses on the left and right hand sides of the equation.
Now, you can understand why it is very convenient to express masses in
the units of energy.\\
Composing equations like (\ref{nucl_reactions.3he3he}), we should
always check the superscripts and subscripts of the nuclei in order to
have the same number of nucleons and the same charge on both sides of
the equation. In the above example, we have six nucleons and the
charge $+4$ in both the initial and final states of the reaction. To
make the checking of nucleon number and charge conservation easier,
sometimes the proton and neutron are denoted with superscripts and
subscripts as well, namely, $^1_1p$ and $^1_0n$. In this case, all we
need is to check that sum of superscripts and sum of subscripts are
the same on both sides of the equation.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Detectors}
\label{nucl_detectors}
How can we observe such tiny tiny things as protons and
$\alpha$-particles? There is no microscope that would be able to
discern them. From the very beginning of the sub-atomic era,
scientists have been working on the development of special instruments
that are called particle detectors. These devices enable us either to
register the mere fact that certain particle has passed through
certain point in space or to observe the trace of its path (the
trajectory). Actually, this is as good as watching the
particle. Although the particle sizes are awfully small, when passing
through some substances, they leave behind visible traces of tens of
centimeters in length. By measuring the curvature of the trajectory of
a particle deflected in electric or magnetic field, a physicist can
determine the charge and mass of the particle and thus can identify
it.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Geiger counter}
\label{nucl_detectors_geiger}
The most familiar device for registering charged particles is the
Geiger counter. It cannot tell you anything about the particle except
the fact that it has passed through the counter. The counter consists
of a thin metal cylinder filled with gas. A wire electrode runs along
the center of the tube and is kept at a high voltage ($\sim 2000$\,V)
relative to the cylinder. When a particle passes through the tube, it
causes ionization of the gas atoms and thus an electric discharge
between the cylinder and the wire. The electric pulse can be counted
by a computer or made to produce a ``click'' in a loudspeaker. The
number of counts per second tells us about intensity of the radiation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fluorescent screen}
\label{nucl_detectors_screen}
The very first detector was the fluorescent screen. When a charged
particle hits the screen, a human eye can discern a flash of light at
the point of impact. In fact, we all use this kind of detectors every
day when watching TV of looking at a computer (if it does not have an
LCD screen of course). Indeed, the images on the screens of their
electron-ray tubes are formed by the accelerated electrons.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Photo-emulsion}
\label{nucl_detectors_emulsion}
Another type of particle detector, dating back to Becquerel, is the
nuclear photographic emulsion. Passage of charged particles is
recorded in the emulsion in the same way that ordinary black and white
photographic film records a picture. The only difference is that
nuclear photoemulsion is made rather thick in order to catch a
significant part of the particle path. After the developing, a
permanent record of the charged particle trajectory is available.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ Wilson's chamber}
\label{nucl_detectors_wilson}
In the fields of sub-atomic physics and nuclear physics, Wilson's cloud
chamber is the most fundamental device to observe the trajectories of
particles. Its basic principle was discovered by C. T. R. Wilson
in 1897, and it was put to the practical use in 1911.\\
The top and the side of the chamber are covered with round glasses of
several centimeters in diameter. At the bottom of the chamber, a
piston is placed. The air filled in the chamber is saturated with
vapor of water. When pulling down the piston quickly, the volume of
the chamber would be expanded and the temperature goes down. As a
result, the air inside would be supersaturated with the vapor. If a
fast moving charged particle enters the chamber when it is in such a
supersaturated state, the vapor of water would condense along the line
of the ions generated by the particle, which is the path of the
particle. Thus we can observe the trace, and also take a
photograph. To make clear the trace, a light is sometimes illuminated
from the side. When placing the cloud chamber in a magnetic field, we
can obtain various informations about the charged particle by
measuring the curvature of the trace and other data. The bubble
chamber and the spark chamber have taken place of the cloud chamber
which is nowadays used only for the educational purposes. Wilson's
cloud chamber has however played a very important role in the history
of physics.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Bubble chamber}
\label{nucl_detectors_bubble}
Bubble chamber is a particle detector of major importance during the
initial years of high-energy physics. The bubble chamber has produced
a wealth of physics from about 1955 well into the 1970s. It is based
on the principle of bubble formation in a liquid heated above its
boiling point, which is then suddenly expanded, starting boiling where
passing charged particles have ionized the atoms of the liquid. The
technique was honoured by the Nobel prize award to D. Glaser in 1960.
Even today, bubble chamber photographs provide the aesthetically most
appealing visualization of subnuclear collisions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spark chamber}
\label{nucl_detectors_spark}
Spark chamber is a historic device using electric discharges over a
gap between two electrodes with large potential difference, to render
passing particles visible. Sparks occurred where the gas had been
ionized. Most often, multiple short gaps were used, but wide-gap
chambers with gaps up to 40 cm were also built. The spark chamber is
still of great scientific value in that it remains relatively simple
and cheap to build as well as enabling an observer to view the paths
of charged particles.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nuclear energy}
\label{nucl_energy}
Nuclei can produce energy via two different types of reactions,
namely, {\it fission} and {\it fusion} reactions.
Fission is a break up of a nucleus in two or more pieces (smaller
nuclei). Fusion is the opposite process: Formation of a bigger nucleus
from two small nuclei.\\
A question may arise: How two opposite processes can both produce
energy? Can we make an inexhaustible souce of energy by breaking up
and then fusing the same nuclei? Of cousre not! The energy conservation
law cannot be circumvented in no way. When speaking about fusion and
fission, we speak about different ranges of nuclei. Energy can only be
released when either light nuclei fuse or heavy nuclei fission.\\
To understand why this is so, let us recollect that for releasing
energy the mass of initial nuclei must be greater than the mass of the
products of a nuclear reaction. The mass difference is transformed
into the released energy. And why the product nuclei can loose some
mass as compared to the initial nuclei? Because they are more tightly
bound, i.e. their binding energies are lager.\\
Fig.~\ref{nucl_energy_be.fig} shows the dependence of the binding
energy $B$ per nucleon on the number $A$ of nucleons constituting a
nucleus. As you see, the curve reaches the maximum value of
$\sim9$\,MeV per nucleon at around $A\sim50$. The nuclei with such
number of nucleons cannot produce energy neither through fusion nor
through fission. They are kind of ``ashes'' and cannot serve as a
fuel. In contrast to them, very light nuclei, when fused with each
other, make more tightly bound products as well as very heavy nuclei
do when split up in lighter fragments.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure nucl_energy_be.fig %%%%%%%%%%%%%%%%
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\caption{\sf Binding energy per nucleon.}
\label{nucl_energy_be.fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In fission processes, which were discovered and used first, a heavy
nucleus like, for example, uranium or plutonium, splits up in two
fragments which are both positively charged. These fragments repel
each other by an electric force and move apart at a high speed,
distributing their kinetic energy in the surrounding material.\\
In fusion reactions everything goes in the opposite direction. Very
light nuclei, like hydrogen or helium isotopes, when approaching each
other to a distance of a few fm (1\,fm = 10$^{-13}$\,cm), experience
strong attraction which overpowers their Coulomb (that is electric)
repulsion. As a result the two nuclei fuse into a single nucleus. They
collapse with extremely high speeds towards each other. To form a
stable nucleus they must get rid of the excessive energy. This energy
is emitted by ejecting a neutron or a photon.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Nuclear reactors}
\label{nucl_energy_reactors}
Since the discovery of radioactivity it was known that heavy nuclei
release energy in the processes of spontaneous decay. This process,
however, is rather slow and cannot be influenced (speed up or slow
down) by humans and therefore could not be effectively used for
large-scale energy production. Nonetheless, it is ideal for feeding
the devices that must work autonomously in remote places for a long
time and do not require much energy. For this, heat from the
spontaneous-decays can be converted into electric power in a
radioisotope thermoelectric generator. These generators have been
used to power space probes and some lighthouses built by Russian
engineers. Much more effective way of using nuclear energy is based
on another type of nuclear decay which is considered next.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Chain reaction}
\label{nucl_energy_reactors_chain}
The discovery that opened up the era of nuclear energy was made in
1939 by German physicists O. Hahn, L. Meitner, F Strassmann, and
O. Frisch. They found that a uranium nucleus, after absorbing a
neutron, splits into two fragments. This was not a spontaneous but
induced fission
\begin{equation}
\label{nucl_energy.n235U}
n+ {}^{235}_{\phantom{0}92}{\rm U} \longrightarrow
{}^{140}_{\phantom{0}54}{\rm Xe} +
{}^{94}_{38}{\rm Sr}
+ n + n + 185\,{\rm MeV}
\end{equation}
that released $\sim185$\,MeV of energy as well as two neutrons which
could cause similar reactions on surrounding nuclei. The fact that
instead of one initial neutron, in the reaction
(\ref{nucl_energy.n235U}) we obtain two neutrons, is crucial. This
gives us the possibility to make the so-called {\it chain reaction}
schematically shown in Fig. \ref{nucl_energy_chain.fig}.\\
%%%%%%%%%%%%%%%%%%%%%%%% Fig. {nucl_energy_chain.fig} %%%%%%%%%%%%
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\caption{\sf Chain reaction on uranium nuclei.}
\label{nucl_energy_chain.fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In such process, one neutron breaks one heavy nucleus, the two
released neutrons break two more heavy nuclei and produce four
neutrons which, in turn, can break another four nuclei and so on.
This process develops extremely fast. In a split of a second a huge
amount of energy can be released, which means explosion. In fact, this
is how the so-called atomic bomb works.\\
Can we control the development of the chain reaction? Yes we can! This
is done in nuclear reactors that produce energy for our use. How can
it be done?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Critical mass}
\label{nucl_energy_reactors_critical}
First of all, if the piece of material containing fissile nuclei is
too small, some neutrons may reach its surface and escape without
causing further fissions. For each type of fissile material there is
therefore a minimal mass of a sample that can support explosive chain
reaction. It is called the {\it critical mass}. For example, the
critical mass of ${}^{235}_{\phantom{0}92}{\rm U}$ is approximately
50\,kg. If the mass is below the critical value, nuclear explosion is
not possible, but the energy is still released and the sample becomes
hot. The closer mass is to its critical value, the more energy is
released and more intensive is the neutron radiation from the
sample.\\
The criticality of a sample (i.e. its closeness to the critical state)
can be reduced by changing its geometry (making its surface bigger) or
by putting inside it some other material (boron or cadmium) that is
able to absorb neutrons. On the other hand, the criticality can be
increased by putting neutron reflectors around the sample. These
reflectors work like mirrors from which the escaped neutrons bounce
back into the sample. Thus, moving in and out the absorbing material
and reflectors, we can keep the sample close to the critical state.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{How a nuclear reactor works}
\label{nucl_energy_reactors_how}
In a typical nuclear reactor, the fuel is not in one piece, but in the
form of several hundred vertical rods, like a brush. Another system of
rods that contain a neutron absorbing material (control rods) can move
up and down in between the fuel rods. When totally in, the control
rods absorb so many neutrons, that the reactor is shut down. To start
the reactor, operator gradually moves the control rods up. In an
emergency situation they are dropped down automatically.\\
To collect the energy, water flows through the reactor core. It
becomes extremely hot and goes to a steam generator. There, the heat
passes to water in a secondary circuit that becomes steam for use
outside the reactor enclosure for rotating turbines that generate
electricity.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Nuclear power in South Africa}
\label{nucl_energy_reactors_SA}
By 2004 South Africa had only one commercial nuclear reactor
supplying power into the national grid. It works in Koeberg located
30\,km north of Cape Town. A small research reactor was also operated
at Pelindaba as part of the nuclear weapons program, but was
dismantled.\\
Koeberg Nuclear Power station is a uranium Pressurized Water Reactor
(PWR). In such a reactor, the primary coolant loop is pressurised so
the water does not boil, and heat exchangers, called steam generators,
are used to transmit heat to a secondary coolant which is allowed to
boil to produce steam. To remove as much heat as possible, the water
temperature in the primary loop is allowed to rise up to about
300\,$^{\circ}$C which requires the pressure of 150 atmospheres (to
keep water from boiling).\\
The Koeberg power station has the largest turbine generators in the
southern hemisphere and produces $\sim$10000\,MWh of electric energy.
Construction of Koeberg began in 1976 and two of its Units were
commissioned in 1984-1985. Since then, the plant has been in more or
less continuous operation and there have been no serious incidents.\\
Eskom that operates this power station, may be the current technology
leader. It is developing a new type of nuclear reactor, a modular
pebble-bed reactor (PBMR). In contrast to traditional nuclear
reactors, in this new type of reactors the fuel is not assembled in
the form of rods. The uranium, thorium or plutonium fuels are in
oxides (ceramic form) contained within spherical pebbles made of
pyrolitic graphite. The pebbles, having a size of a tennis ball, are
in a bin or can. An inert gas, helium, nitrogen or carbon dioxide,
circulates through the spaces between the fuel pebbles. This carries
heat away from the reactor.\\
Ideally, the heated gas is run directly through a turbine. However
since the gas from the primary coolant can be made radioactive by the
neutrons in the reactor, usually it is brought to a heat exchanger,
where it heats another gas, or steam.\\
The primary advantage of pebble-bed reactors is that they can be
designed to be inherently safe. When a pebble-bed reactor gets
hotter, the more rapid motion of the atoms in the fuel increases the
probability of neutron capture by $^{238}_{\phantom{0}92}$U isotopes
through an effect known as Doppler broadening. This isotope does not
split up after capturing a neutron. This reduces the number of
neutrons available to cause $^{235}_{\phantom{0}92}$U fission,
reducing the power output by the reactor. This natural negative
feedback places an inherent upper limit on the temperature of the fuel
without any operator intervention.\\
The reactor is cooled by an inert, fireproof gas, so it cannot have a
steam explosion as a water reactor can.\\
A pebble-bed reactor thus can have all of its supporting machinery
fail, and the reactor will not crack, melt, explode or spew hazardous
wastes. It simply goes up to a designed "idle" temperature, and stays
there. In that state, the reactor vessel radiates heat, but the vessel
and fuel spheres remain intact and undamaged. The machinery can be
repaired or the fuel can be removed.\\
A large advantage of the pebble bed reactor over a conventional
water reactor is that they operate at higher temperatures. The
reactor can directly heat fluids for low pressure gas turbines. The
high temperatures permit systems to get more mechanical energy from
the same amount of thermal energy.\\
Another advantage is that fuel pebbles for different fuels might be
used in the same basic design of reactor (though perhaps not at the
same time). Proponents claim that some kinds of pebble-bed reactors
should be able to use thorium, plutonium and natural unenriched
Uranium, as well as the customary enriched uranium. One of the
projects in progress is to develop pebbles and reactors that use the
plutonium from surplus or expired nuclear explosives.\\
On June 25, 2003, the South African Republic's Department of
Environmental Affairs and Tourism approved ESKOM's prototype 110\,MW
pebble-bed modular reactor for Koeberg. Eskom also has approval for a
pebble-bed fuel production plant in Pelindaba. The uranium for this
fuel is to be imported from Russia. If the trial is successful, Eskom
says it will build up to ten local PBMR plants on South Africa's
seacoast. Eskom also wants to export up to 20 PBMR plants per
year. The estimated export revenue is 8 billion rand a year, and could
employ about 57000 people.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fusion energy}
\label{nucl_energy_fusion}
For a given mass of fuel, a fusion reaction like
\begin{equation}
\label{nucl_energy_fusion_dt}
{}^2_1{\rm H} + {}^3_1{\rm H} \longrightarrow
{}^4_2{\rm He} + n + 17.59\,{\rm MeV}\ .
\end{equation}
yield several times more energy than a fission reaction. This is clear
from the curve given in Fig.~\ref{nucl_energy_be.fig}. Indeed, a
change of the binding energy (per nucleon) is much more
significant for a fusion reaction than for a fission reaction. Fusion
is, therefore, a much more powerful source of energy.
For example, 10\,g of Deuterium which can be extracted from 500
litres of water and 15\,g of Tritium produced from 30\,g of Lithium would
give enough fuel for the lifetime electricity needs of an average
person in an industrialised country.\\
But this is not the only reason why fusion attracted so much attention
from physicists. Another, more fundamental, reason is that the fusion
reactions were responsible for the synthesis of the initial amount of
light elements at primordial times when the universe was
created. Furthermore, the synthesis of nuclei continues inside the
stars where the fusion reactions produce all the energy which reaches
us in the form of light.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Thermonuclear reactions}
\label{nucl_energy_fusion_thermo}
If fusion is so advantageous, why is it not used instead of fission
reactors? The problem is in the electric repulsion of the
nuclei. Before the nuclei on the left hand side of Eq.
(\ref{nucl_energy_fusion_dt}) can fuse, we have to bring them somehow
close to each other to a distance of $\sim10^{-13}$\,cm. This is not
an easy task! They both are positively charged and ``refuse'' to
approach each other.\\
What we can do is to make a mixture of the atoms containing such
nuclei and heat it up. At high temperatures the atoms move very fast.
They fiercely collide and loose all the electrons. The mixture becomes
{\it plasma}, i.e. a mixture of bare nuclei and free moving electrons.
If the temperature is high enough, the colliding nuclei can overcome the
electric repulsion and approach each other to a fusion distance.\\
When the nuclei fuse, they release much more energy than was spent to
heat up the plasma. Thus the initial energy ``investment'' pays off.
The typical temperature needed to ignite the reaction of the type
(\ref{nucl_energy_fusion_dt}) is extremely high. In fact, it is the
same temperature that our sun has in its center, namely, $\sim$15
million degrees. This is why the reactions
(\ref{nucl_reactions.3he3he}), (\ref{nucl_energy_fusion_dt}), and the
like are called {\it thermonuclear reactions}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Human-made thermonuclear reactions}
\label{nucl_energy_fusion_made_thermo}
The same as with fission reactions, the first application of
thermonuclear reactions was in weapons, namely, in the hydrogen bomb,
where fusion is ignited by the explosion of an ordinary (fission)
plutonium bomb which heats up the fuel to solar temperatures.\\
In an attempt to make a controllable fusion, people encounter the
problem of holding the plasma. It is relatively easy to achieve a high
temperature (with laser pulses, for example). But as soon as
plasma touches the walls of the container, it immediately cools down.
To keep it from touching the walls, various ingenious methods are
tried, such as strong magnetic field and laser beams directed to
plasma from all sides. In spite of all efforts and ingenious tricks,
all such attempts till now have failed. Most probably this
straightforward approach to controllable fusion is doomed because one
has to hold in hands a ``piece of burning sun''.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Cold fusion}
\label{nucl_energy_fusion_cold}
To visualize the struggle of the nuclei approaching each other,
imagine yourself pushing a metallic ball towards the top of a slope
shown in Fig. \ref{nucl_energy_fusion_cold_veff.fig}. The more kinetic
energy you give to the ball, the higher it can climb. Your purpose is
to make it fall into the narrow well that is behind the barrier.\\
%%%%%%%%%%%%%%%%%%%%%%%% FIGURE: COULOMB BARRIER %%%%%%%%%%%%%%%%%
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\end{picture}
\caption{\sf Effective nucleus--nucleus potential as a function of the
separation between the nuclei.}
\label{nucl_energy_fusion_cold_veff.fig}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In fact, the curve in Fig. \ref{nucl_energy_fusion_cold_veff.fig}
shows the dependence of relative potential energy $V_{\rm eff}$
between two nuclei on the distance $R$ separating them. The deep
narrow well corresponds to the strong short-range attraction, and the
$\sim1/R$ barrier represents the Coulomb (electric) repulsion. The
nuclei need to overcome this barrier in order to ``touch'' each other
and fuse, i.e. to fall into the narrow and deep potential well. One
way to achieve this is to give them enough kinetic energy, which means
to rise the temperature. However, there is another way based on the
quantum laws.\\
As you remember, when discussing the motion of the electron inside an
atom (see Sec. \ref{nucl_what_is}), we said that it formed a ``cloud''
of probability around the nucleus. The density of this cloud
diminishes at very short and very long distances but never disappears
completely. This means that we can find the electron even inside the
nucleus though with a rather small probability.\\
The nuclei moving towards each other, being microscopic objects, obey
the quantum laws as well. The probability density for finding one
nucleus at a distance $R$ from another one also forms a cloud. This
density is non-zero even under the barrier and on the other side of
the barrier. This means that, in contrast to classical objects,
quantum particles, like nuclei, can penetrate through potential
barriers even if they do not have enough energy to go over it! This is
called the {\it tunneling effect}.\\
The tunneling probability strongly depends on thickness of the
barrier. Therefore, instead of lifting the nuclei against the barrier
(which means rising the temperature), we can try to make the barrier
itself thinner or to keep them close to the barrier for such a long
time that even a low penetration probability would be realized.\\
How can this be done? The idea is to put the nuclei we want to fuse,
inside a molecule where they can stay close to each other for a long
time. Furthermore, in a molecule, the Coulomb barrier becomes thinner
because of electron screening. In this way fusion may proceed even at
room temperature.\\
This idea of {\it cold fusion} was originally (in 1947) discussed by
F. C. Frank and (in 1948) put forward by A. D. Sakharov, the
``father'' of Russian hydrogen bomb, who at the latest stages of his
career was worldwide known as a prominent human rights activist and a
winner of the Nobel Prize for Peace. When working on the bomb project,
he initiated research into peaceful applications of nuclear energy and
suggested the fusion of two hydrogen isotopes via the reaction
(\ref{nucl_energy_fusion_dt}) by forming a molecule of them where one of
the electrons is replaced by a muon.\\
The muon is an elementary particle (see Sec. \ref{nucl_particles}),
which has the same characteristics as an electron. The only difference
between them is that the muon is 200 times heavier than the
electron. In other words, a muon is a heavy electron. What will happen
if we make a muonic atom of hydrogen, that is a bound state of a
proton and a muon? Due to its large mass the muon would be very close
to the proton and the size of such atom would be 200 times smaller
than that of an ordinary atom. This is clearly seen from the formula
for the atomic Bohr radius
$$
R_{\mbox{\tiny Bohr}}=\frac{\hbar^2}{me^2}\ ,
$$
where the mass is in the denominator.\\
Now, what happens if we make a muonic molecule? It will also be 200
times smaller than an ordinary molecule. The Coulomb barrier will be
200 times thinner and the nuclei 200 times closer to each other. This
is just what we need! Speaking in terms of the effective
nucleus--nucleus potential shown in
Fig. \ref{nucl_energy_fusion_cold_veff.fig}, we can say that the muon
modifies this potential in such a way that a second minimum
appears. Such a modified potential is (schematically) shown in
Fig.~\ref{nucl_energy_fusion_cold_Vmol.fig}.\\
%%%%%%%%%%%%%%%%%%%%%%%% FIGURE: BARRIER IN A MOLECULE %%%%%%%%%%%%%%%%
\begin{figure}[h!tb]
\begin{center}
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\caption
{\sf Effective nucleus--nucleus potential (thick curve) for nuclei
confined in a molecule. Thin curve shows the corresponding
distribution of the probability for finding the nuclei at a given
distance from each other.}
\label{nucl_energy_fusion_cold_Vmol.fig}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The molecule is a bound state in the shallow but wide minimum of this
potential. Most of the time, the nuclei are at the distance
corresponding to the maximum of the probability density distribution
(shown by the thin curve). Observe that this density is not zero under
the barrier (though is rather small) and even at $R=0$. This means
that the system can (with a small probability) jump from the shallow
well into the deep well through the barrier, i.e. can tunnel and
fuse.\\
Unfortunately, the muon is not a stable particle. Its lifetime is only
$\sim10^{-6}$\,sec. This means that a muonic molecule cannot exist
longer than 1 microsecond. As a matter of fact, from a quantum
mechanical point of view, this is quite a long interval.\\
The quantum mechanical wave function (that describes the probability
density) oscillates with a frequency which is proportional to the
energy of the system. With a typical binding energy of a muonic
molecule of 300\,eV this frequency is $\sim 10^{17}\,{\rm
s}^{-1}$. This means that the particle hits the barrier with this
frequency and during 1 microsecond it makes $10^{11}$ attempts to jump
through it. The calculations show that the penetration probability is
$\sim 10^{-7}$. Therefore, during 1 microsecond nuclei can penetrate
through the barrier 10000 times and fusion can happen much faster than
the decay rate of the muon.\\
Cold fusion via the formation of muonic molecules was done in many
laboratories, but unfortunately, it cannot solve the problem of energy
production for our needs. The obstacle is the negative efficiency,
{\em i.e.} to make muonic cold fusion we have to spend more energy
than it produces. The reason is that muons do not exist like protons
or electrons. We have to produce them in accelerators. This takes a
lot of energy.\\
Actually, the muon serves as a catalyst for the fusion reaction. After
helping one pair of nuclei to fuse, the muon is liberated from the
molecule and can form another molecule, and so on. It was estimated
that the efficiency of the energy production would be positive only if
each muon ignited at least 1000 fusion events. Experimentalists tried
their best, but by now the record number is only 150 fusion events per
muon. This is too few. The main reason why the muon does not catalyze
more reactions is that it is eventually trapped by a $^4$He nucleus
which is a by-product of fusion. Helium captures the muon into an
atomic orbit with large binding energy, and it cannot escape.\\
Nonetheless, the research in the field of cold fusion continues. There
are some other ideas of how to keep nuclei close to each other. One of
them is to put the nuclei inside a crystal. Another way out is to
increase the penetration probability by using molecules with special
properties, namely, those that have quantum states with almost the
same energies as the excited states on the compound nucleus. Scientists
try all possibilities since the energy demands of mankind grow
continuously and therefore the stakes in this quest are high.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Elementary particles}
\label{nucl_particles}
In our quest for the elementary building blocks of the universe, we
delved inside atomic nucleus and found that it is composed of protons
and neutrons. Are the three particles, $e$, $p$, and $n$, the blocks
we are looking for? The answer is ``no''. Even before the structure
of the atom was understood, Becquerel discovered the redioactivity
(see Sec. \ref{nucl_decays_discovery}) that afterwards puzzled
physicists and forced them to look deeper, i.e. inside protons and
neutrons.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\beta$ decay}
\label{nucl_particles_beta}
%------------------------------------------------------
Among the three types of radioactivity, the $\alpha$ and $\gamma$ rays
were easily explained. The emission of $\alpha$ particle is kind of
fission reaction, when an initial nucleus spontaneously decays in two
fragments one of which is the nucleus $^4_2$He (i.e. $\alpha$ particle).
The $\gamma$ rays are just electromagnetic quanta emitted by a nuclear
system when it transits from one quantum state to another (the same
like an atom emits light).\\
The $\beta$ rays posed the puzzle. On the one hand, they are just
electrons and you may think that it looks simple. But on the other
hand, they are not the electrons from the atomic shell. It was found
that they come from inside the nucleus! After the $\beta$-decay, the
charge of the nucleus increases in one unit,
$$
^A_Z\left(\mbox{\sf parent nucleus}\right)\,\longrightarrow\,
^{\phantom{1+}A}_{Z+1}\left(\mbox{\sf daughter nucleus}\right)+e\ ,
$$
which is in accordance with the charge conservation law.\\
There was another puzzle associated with the $\beta$ decay: The emitted
electrons did not have a certain energy. Measuring their kinetic
energies, you could find very fast and very slow electrons as well as
the electrons with all intermediate speeds. How could identical parent
nuclei, after loosing different amount of energy, become identical
daughter nuclei. May be energy is not conserving in the quantum world?
The fact was so astonishing that even Niels Bohr put forward the idea
of statistical nature of the energy conservation law.\\
To explain the first puzzle, it was naively suggested that neutron is
a bound state of proton and electron. At that time, physicists
believed that if something is emitted from an object, this something
must be present inside that object before the emission. They could not
imagine that a particle could be created from vacuum.\\
The naive $(pe)$ model of the neutron contradicted the facts. Indeed,
it was known already that the $pe$ bound state is the hydrogen
atom. Neutron is much smaller than the atom. Therefore, it would be
unusually tight binding, and perhaps with something elese involved
that keeps the size small. By the way, this ``something elese'' could
also save the energy conservation law. In 1930, Wolfgang Pauli
suggested that in addition to the electron, the $\beta$ decay involves
another particle, $\nu$, that is emitted along with the electron and
carries away part of the energy. For example,
\begin{equation}
\label{nucl_particles_beta_example}
^{234}_{\phantom{0}90}{\rm Th}\,
\longrightarrow\,
^{234}_{\phantom{0}91}{\rm Pa} + e^{-} + \bar{\nu}\ .
\end{equation}
This additional particle was called {\it neutrino} (in Italian the
word ``neutrino'' means small neutron). The neutrino is electrically
neutral, has extremely small mass (maybe even zero, which is still a
question in 2004) and very weakly interacts with matter. This is why
it was not detected experimentally till 1956. The ``bar'' over $\nu$
in Eq. (\ref{nucl_particles_beta_example}) means that in this reaction
actually the anti-neutrino is emitted (see the discussion on
anti-particles further down in Sec. \ref{nucl_particles_physics}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Particle physics}
\label{nucl_particles_physics}
%------------------------------------------------------
In an attempt to explain the $\beta$ decay and to understand internal
structure of the neutron a new branch of physics was born, the {\it
particle physics}. The only way to explore the structure of sub-atomic
particles is to strike them with other particles in order to knock out
their ``constituent'' parts. The simple logic says: The more powerful
the impact, the smaller parts can be knocked out.\\
At the beginning the only source of energetic particles to strike
other particles were the cosmic rays. Earth is constantly bombarded by
all sort of particles coming from the outer space. Atmosphere protects
us from most of them, but many still reach the ground.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Antiparticles}
\label{nucl_particles_physics_anti}
%------------------------------------------------------
In 1932, studying the cosmic rays with a bubble chamber, Carl Anderson
made a photograph of two symmetrical tracks of charged particles. The
measurements of the track curvatures showed that one track belonged to
an electron and the other was made by a particle having the same mass
and equal but positive charge. These particles were created when a
cosmic $\gamma$ quantum of a high energy collided with a nucleus.\\
The discovered particle was called {\it positron} and denoted as $e^+$
to distinguish it from the electron, which sometimes is denoted as
$e^-$. It was the first antiparticle discovered. Later, it was found
that every particle has its ``mirror reflection'', the
antiparticle. To denote an antiparticle, it is used ``bar'' over a
particle symbol. For example, $\bar{p}$ is the anti-proton, which has
the same mass as an ordinary proton but a negative charge.\\
When a particle collides with its ``mirror reflection'', they
annihilate, i.e. they burn out completely. In this collision, all
their mass is transformed into electromagnetic energy in the form of
$\gamma$ quanta. For example, if an electron collides with a positron,
the following reaction may take place
\begin{equation}
\label{nucl_particles_physics_anti_ee}
e^-+e^+\,
\longrightarrow\,\gamma+\gamma\ ,
\end{equation}
where two photons are needed to conserve the total momentum of the
system.\\
In principle, stable antimatter can exist. For example, the pair of
$\bar{p}$ and $e^+$ can form an atom of anti-hydrogen with exactly the
same energy states as the ordinary hydrogen. Experimentally,
atoms of anti-helium were obtained. The problem with them is that,
surrounded by ordinary matter, they cannot live long. Colliding with
ordinary atoms, they annihilate very fast.\\
There are speculations that our universe should be symmetric with
respect to particles and antiparticles. Indeed, why should preference
be given to matter and not to anti-matter? This implies that somewhere
very far, there must be equal amount of anti-matter, i.e. anti-universe.
Can you imagine what happens if they meet?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Muon, mesons, and the others}
\label{nucl_particles_physics_muon}
%------------------------------------------------------
In yet another cosmic-ray experiment a particle having the same
properties as the electron but $\sim$207 times heavier, was discovered
in 1935. It was given the name {\it muon} and the symbol $\mu$. For a
long time it remained ``unnecessary'' particle in the picture of the
world. Only the modern theories harmonically included the muon as a
constituent part of matter (see Sec \ref{nucl_particles_quarks}).\\
The same inexhaustible cosmic rays revealed the $\pi$ and $K$ mesons
in 1947. The $\pi$ mesons (or simply {\it pions}) were theoretically
predicted twelve years before by Yukawa, as the mediators of the
strong forces between nucleons. The $K$ mesons, however, were
unexpected. Furthermore, they showed very strange behaviour. They were
easily created only in pairs. The probability of the inverse process
(i.e. their decay) was $10^{13}$ times lower than the probability of
their creation.\\
It was suggested that these particles possess a new type of charge,
the {\it strangeness}, which is conserving in the strong
interactions. When a pair of such particles is created, one of them
has strangeness $+1$ and the other $-1$, so the total strangeness
remains zero. When decaying, they act individually and therefore the
strangeness is not conserving. According to the suggestion, this is
only possible through the weak interactions that are much weaker than
the strong interactions (see Sec. \ref{nucl_particles_forces}) and
thus the decay probability is much lower.\\
The golden age of particle physics began in 1950-s with the advent of
particle accelerators, the machines that produced beams of electrons
or protons with high kinetic energy. Having such beams available,
experimentalists can plan the experiment and repeat it, while with the
cosmic rays they were at the mercy of chance. When the accelerators
became the main tool of exploration, the particle physics acquired its
second name, the {\it high energy physics}.\\
During the last half a century, experimentalists discovered so many
new particles (few of them are listed in Table
\ref{tabl.nucl_particles_physics_muon}) that it became obvious that
they cannot all be elementary. When colliding with each other, they
produce some other particles. Mutual transformations of the particles
is their main property.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1.0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[!h!t!b]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
{\sf family} & {\sf particle} & {\sf symbol} & {\sf mass} (MeV) &
{\sf Lifetime} $T_{1/2}$ (s)\\
\hline
photon & photon & $\gamma$ & 0 & stable\\
\hline
& electron & $e^-$, $e^+$ & 0.511 & stable \\
\cline{2-5}
& muon & $\mu^-$, $\mu^+$ & 105.7 & $2.2\times10^{-6}$ \\
\cline{2-5}
leptons & tau & $\tau^-$, $\tau^+$ & 1777 & $10^{-13}$ \\
\cline{2-5}
& electron neutrino & $\nu_e$ & $\sim0$ & stable \\
\cline{2-5}
& muon neutrino & $\nu_{\mu}$ & $\sim0$ & stable \\
\cline{2-5}
& tau neutrino & $\nu_{\tau}$ & $\sim0$ & stable \\
\hline
& pion & $\pi^+$, $\pi^-$ & 139.6 & $2.6\times10^{-8}$\\
& pion & $\pi^0$ & 135.0 & $0.8\times10^{-16}$\\
\cline{2-5}
& kaon & $K^+$, $K^-$ & 493.7 & $1.2\times10^{-8}$\\
& kaon & $K^0_S$ & 497.7 & $0.9\times10^{-10}$\\
hadrons & kaon & $K^0_L$ & 497.7 & $5.2\times10^{-8}$\\
\cline{2-5}
& eta meson & $\eta^0$ & 548.8 & $10^{-18}$\\
\cline{2-5}
& proton & $p$ & 938.3 & stable\\
\cline{2-5}
& neutron & $n$ & 939.6 & 900 \\
\cline{2-5}
& lambda & $\Lambda^0$ & 1116 & $2.6\times10^{-10}$\\
\cline{2-5}
& sigma & $\Sigma^+$ & 1189 & $0.8\times10^{-10}$\\
& sigma & $\Sigma^0$ & 1192 & $6\times10^{-20}$\\
& sigma & $\Sigma^-$ & 1197 & $1.5\times10^{-10}$\\
\cline{2-5}
& omega & $\Omega^-$, $\Omega^+$& 1672 & $0.8\times10^{-10}$\\
\hline
\end{tabular}
\caption{\sf Few representatives of different particle families.}
\label{tabl.nucl_particles_physics_muon}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Physicists faced the problem of particle classification similar to the
problems of classification of animals, plants, and chemical elements.
The first approach was very simple. The particles were divided in four
groups according to their mass: {\it leptons} (light particles, like
electron), {\it mesons} (intermediate mass, like pion), {\it baryons}
(heavy particles, like proton or neutron), and {\it hyperons} (very
heavy particles).\\
Then it was realized that it would be more logical to divide the
particles in three families according to their ability to interact via
weak, electromagnetic, and strong forces (in addition to that, all
particles experience gravitational attraction towards each
other). Except for the gravitational interaction, the photon ($\gamma$
quantum) participates only in electromagnetic interactions, the
leptons take part in both weak and electromagnetic interactions, and
{\it hadrons} are able to interact via all forces of nature (see
Sec. \ref{nucl_particles_forces}).\\
In addition to conservation of the strangeness, several other
conservation laws were discovered. For example, number of leptons is
conserving. This is why in the reaction
(\ref{nucl_particles_beta_example}) we have an electron (lepton number
$+1$) and anti-neutrino (lepton number $-1$) in the final
state. Similarly, the number of baryons is conserving in all
reactions.\\
The quest for the constituent parts of the neutron has led us to
something unexpected. We found that there are several hundreds of
different particles that can be ``knocked out'' of the neutron but
none of them are its parts. Actually, the neutron itself can be
knocked out of some of them! What a mess! Further efforts of
experimentalists could not find an order, which was finally discovered
by theoreticians who introduced the notion of {\it quarks}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Quarks and leptons}
\label{nucl_particles_quarks}
%------------------------------------------------------
While experimentalists seemed to be lost in the maze, the
theoreticians groped for the way out. Using an extremely complicated
mathematical technique, they managed to group the hadrons in such
families which implied that all known (and yet unknown) hadrons are
build of only six types of particles with fractional charges.
The main credit for this (in the form of Nobel Prize) was given to
M. Gell-Mann and G. Zweig.\\
At first, they considered a subset of the hadrons and developed a
theory with only three types of such truly elementary particles. When
Murray Gell-Mann thought of the name for them, he came across the book
"Finnegan's Wake" by James Joyce. The line "Three quarks for Mister
Mark..." appeared in that fanciful book (in German, the word ``quark''
means cottage cheese). He needed a name for three particles and this
was the answer. Thus the term {\it quark} was coined.\\
Later, the theory was generalized to include all known particles,
which required six types of quarks. Modern theories require also that
the number of different leptons should be the same as the number of
different quark types. According to these theories, the quarks and
leptons are truly elementary, i.e. they do not have any internal
structure and therefore are of a zero size (pointlike). Thus, the
world is constructed of just twelve types of elementary building
blocks that are given in Table \ref{tabl.nucl_particles_quarks}.
Amazingly enough, the electron that was discovered before all other
particles, more than a century ago, turned out to be one of them!\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1.3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{\sf family} & {\sf elementary particle} & {\sf symbol} & {\sf charge}
& \parbox[c]{13mm}{\sf lepton\\[-1mm] number\\[-3mm]}
& \parbox[c]{13mm}{\sf baryon\\[-1mm] number\\[-3mm]}
& {\sf mass} (MeV)\\
\hline
& electron & $e^-$ & $-1$ & 1 & 0 & 0.511\\
\cline{2-7}
& muon & $\mu^-$ & $-1$ & 1 & 0 & 105.7\\
\cline{2-7}
leptons & tau & $\tau^-$ & $-1$ & 1 & 0 & 1777 \\
\cline{2-7}
& electron neutrino & $\nu_e$ & $0$ & 1 & 0 & $\sim0$\\
\cline{2-7}
& muon neutrino & $\nu_{\mu}$ & $0$ & 1 &0 & $\sim0$ \\
\cline{2-7}
& tau neutrino & $\nu_{\tau}$ & $0$ & 1 &0 & $\sim0$ \\
\hline
& up & $u$ & $+2/3$ & 0 &1/3 & 360\\
\cline{2-7}
& down & $d$ & $-1/3$ & 0 &1/3 & 360\\
\cline{2-7}
quarks & strange & $s$ & $-1/3$ & 0 &1/3 & 1500\\
\cline{2-7}
& charmed & $c$ & $+2/3$ & 0 &1/3 & 540\\
\cline{2-7}
& top (truth) & $t$ & $+2/3$ & 0 &1/3 & 174000\\
\cline{2-7}
& bottom (beauty) & $b$ & $-1/3$ & 0 &1/3 & 5000\\
\hline
\end{tabular}
\caption{\sf Elementary building blocks of the universe.}
\label{tabl.nucl_particles_quarks}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
After Gell-Mann, who used a funny name (quark) for an elementary
particle, the fundamental physics was flooded with such names. For
example, the six quark types are called {\it flavors} (for cottage
cheese, this is appropriate indeed), the three different states in
which each quark can be, are called {\it colors} (red, green, blue),
etc. Modern physics is so complicated and mathematical, that people
working in it, need such kind of jokes to ``spice unsavoury dish with
flavors''. The funny names should not confuse anybody. Elementary
particles do not have any smell, taste, or colour. These terms simply
denote certain properties (similar to electric charge) that do not
exist in human world.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Hadrons}
\label{nucl_particles_quarks_hadrons}
%------------------------------------------------------
There are particles that are able to interact with each other by the
so-called {\it strong forces}. Another name for these forces is {\it
nuclear forces}. They are very strong at short distances ($\sim
10^{-15}$\,m), and very quickly vanish when the distance between the
particles increases. All these particles are called {\it hadrons}. The
protons and neutrons are examples of hadrons.\\
As you remember, we learned about the existence of huge variety of
particles when trying to look inside a nucleon, more particularly, the
neutron. So, what the neutron is made of? Can we get the answer at last,
after learning about the quarks? Yes, we can.\\
According to modern theories, all hadrons are composed of quarks. The
quarks can be combined in groups of two or three. The bound states of
two quarks are called {\it mesons}, and the bound complexes of three
quarks are called {\it baryons}. No other numbers of quarks can form
observable particles\footnote{Recently, experimentalists and
theoreticians started to actively discuss the possibility of the
existence of {\it pentaquarks}, exotic particles that are bound
complexes of five quarks.}.\\
%%%%%%%%%%%%%%%%%%%%%%%% FIGURE: QUARK CONTENT OF HADRONS %%%%%%%%%%%%%%%%
\begin{figure}[h!tb]
\begin{center}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PION
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\caption
{\sf Quark content of the proton, neutron, and $\pi^+$-meson.}
\label{fig.nucl_particles_quarks_hadrons_pn}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Nucleons are baryons and therefore consist of three quarks while the
pion is a meson containing only two quarks, as schematically shown in
Fig. \ref{fig.nucl_particles_quarks_hadrons_pn}. Comparing this
figure with Table \ref{tabl.nucl_particles_quarks}, you can see why
quarks have fractional charges. Counting the total charge of a hadron,
you should not forget that anti-quarks have the opposite charges. The
baryon number for an anti-quark also has the opposite sign
(negative). This is why mesons actually consist of a quark and
anti-quark in order to have total baryon number zero.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Particle reactions}
\label{nucl_particles_quarks_reactions}
%------------------------------------------------------
At the early stages of the particle physics development, in order to
find the constituent parts of various particles, experimentalists
simply collided them and watched the ``fragments''. However, this
straightforward approach led to confusion. For example, the reaction
between the $\pi^-$ meson and proton,
\begin{equation}
\label{nucl_particles_quarks_reactions.pip1}
\pi^- + p\,\longrightarrow\, K^0 + \Lambda^0\ ,
\end{equation}
would suggest (if naively interpreted) that either $K^0$ or $\Lambda^0$
is a constituent part of the nucleon while the pion is incorporated
into the other ``fragment''. On the other hand, the same collision can
knock out different ``fragments'' from the same proton. For example,
\begin{equation}
\label{nucl_particles_quarks_reactions.pip2}
\pi^- + p\,\longrightarrow\, \pi^0 + n\ ,
\end{equation}
which leads to an absurd suggestion that neutron is a constituent part
of proton.\\
The quark model explains all such ``puzzles'' nicely and
logically. Similarly to chemical reactions that are just
rearrangements of atoms, the particle reactions of the type
(\ref{nucl_particles_quarks_reactions.pip1}) and
(\ref{nucl_particles_quarks_reactions.pip2}) are just rearrangements
of the quarks. The only difference is that, in contrast to chemistry
where the number of atoms is not changing, the number of quarks before
the collision is not necessarily equal to their number after the
collision. This is because a quark from one colliding particle can
annihilate with the corresponding antiquark from another
particle. Moreover, if the collision is sufficiently powerful, the
quark-antiquark pairs can be created from vacuum.\\
It is convenient to depict the particle transformations in the form of
the so-called {\it quark flow diagrams}. On such diagrams, the quarks
are represented by lines that may be visualized as the trajectories
showing their movement from the left to the right.\\
%%%%%%%%%%%%%%%%%%%%%%%% FIGURE: QUARK FLOW DIAGRAM 1 %%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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%
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\put(0,0){$\left\{\begin{array}{c}
\phantom{1}\\[-3mm]
\phantom{2}
\end{array}
\right.$}
\end{picture}}
%
\put(9,18.5){\begin{picture}(0,0)%
\put(-5,0){$p$}
\put(0,0){$\left\{\begin{array}{c}
\phantom{1}\\
\phantom{2}
\end{array}
\right.$}
\end{picture}}
%
\put(105,46){\begin{picture}(0,0)%
\put(13,0){$K^0$}
\put(0,0){$\left.\begin{array}{c}
\phantom{1}\\[-3mm]
\phantom{2}
\end{array}
\right\}$}
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%
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\put(13,0){$\Lambda^0$}
\put(0,0){$\left.\begin{array}{c}
\phantom{1}\\
\phantom{2}
\end{array}
\right\}$}
\end{picture}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{picture}}
\end{picture}
\caption
{\sf Quark-flow diagram for the reaction
$\pi^- + p\,\longrightarrow\, K^0 + \Lambda^0$\ .}
\label{fig.nucl_particles_quarks_reactions.pip1}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For example, the diagram given in
Fig. (\ref{fig.nucl_particles_quarks_reactions.pip1}), shows the quark
rearrangement for the reaction
(\ref{nucl_particles_quarks_reactions.pip1}). As you can see, when the
pion collides with proton, its $\bar{u}$ quark annihilates with the
$u$ quark from the proton. At the same time, the $s\bar{s}$ pair is
created from the vacuum. Then, the $\bar{s}$ quark binds with the $d$
quark to form the strange meson $K^0$, while the $s$ quark goes
together with the $ud$ pair as the strange baryon $\Lambda^0$.\\
%%%%%%%%%%%%%%%%%%%%%%%% FIGURE: QUARK FLOW DIAGRAM 2 %%%%%%%%%%%%%%%%
\begin{figure}[h!tb]
\begin{center}
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%%%%%%%%%%% FRAME %%%%%%%%%%%%%
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%
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%
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\put(-8,0){$\pi^-$}
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\phantom{1}\\[-3mm]
\phantom{2}
\end{array}
\right.$}
\end{picture}}
%
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\put(-5,0){$p$}
\put(0,0){$\left\{\begin{array}{c}
\phantom{1}\\
\phantom{2}
\end{array}
\right.$}
\end{picture}}
%
\put(105,46){\begin{picture}(0,0)%
\put(13,0){$\pi^0$}
\put(0,0){$\left.\begin{array}{c}
\phantom{1}\\[-3mm]
\phantom{2}
\end{array}
\right\}$}
\end{picture}}
%
\put(105,18.5){\begin{picture}(0,0)%
\put(13,0){$n$}
\put(0,0){$\left.\begin{array}{c}
\phantom{1}\\
\phantom{2}
\end{array}
\right\}$}
\end{picture}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{picture}}
\end{picture}
\caption
{\sf Quark-flow diagram for the reaction
$\pi^- + p\,\longrightarrow\, \pi^0 + n$\ .}
\label{fig.nucl_particles_quarks_reactions.pip2}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The charge-exchange reaction
(\ref{nucl_particles_quarks_reactions.pip2}) is a more simple
rearrangement process shown in
Fig. \ref{fig.nucl_particles_quarks_reactions.pip2}.
You may wonder why the quark and antiquark of the same flavor in the
$\pi^0$ meson do not annihilate. Yes they do, but not immediately. And
due to this annihilation, the lifetime of $\pi^0$ is 100 million times
shorter than the lifetime of $\pi^{\pm}$ (see Table
\ref{tabl.nucl_particles_physics_muon}).\\
Despite its simplicity, the quark-flow diagram technique is very
powerful method not only for explaining the observed reactions but also
for predicting new reactions that have not yet been seen in
experiments. Knowing the quark content of particles (which is
available in modern Physics Handbooks), you can draw plenty of such
diagrams that will describe possible particle transformations. The
only rule is to keep the lines continuous. They can disappear or
emerge only for a quark-antiquark pair of the same flavor.\\
However, the continuity of the quark lines is valid only for the
processes caused by the strong interaction. Indeed, the $\beta$-decay
of a free neutron (caused by the weak forces),
\begin{equation}
\label{nucl_particles_quarks_reactions.beta}
n\,\longrightarrow\, p+e^-+\bar{\nu}_e\ ,
\end{equation}
as well as the $\beta$-decay of the nuclei, indicate that quarks can
change flavor. In particular, the $\beta$-decay
(\ref{nucl_particles_quarks_reactions.beta}) or
(\ref{nucl_particles_beta_example}) happens because the $d$ quark
transformes into the $u$ quark,
\begin{equation}
\label{nucl_particles_quarks_reactions.ud}
d\,\longrightarrow\, u+e^-+\bar{\nu}_e\ ,
\end{equation}
due to the weak interaction, as shown in Fig.
\ref{fig.nucl_particles_quarks_reactions.beta}
%%%%%%%%%%%%%%%%%%%%%%%% FIGURE: QUARK FLOW DIAGRAM 3 %%%%%%%%%%%%%%%%
\begin{figure}[ht!b]
\begin{center}
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%
\put(17,25){\llap{$d$}}
\put(17,19){\llap{$u$}}
\put(17,13){\llap{$d$}}
%
\put(108,25){\rlap{$u$}}
\put(108,19){\rlap{$u$}}
\put(108,13){\rlap{$d$}}
%
\put(9,18.5){\begin{picture}(0,0)%
\put(-5,0){$n$}
\put(0,0){$\left\{\begin{array}{c}
\phantom{1}\\
\phantom{2}
\end{array}
\right.$}
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%
\put(105,18.5){\begin{picture}(0,0)%
\put(13,0){$p$}
\put(0,0){$\left.\begin{array}{c}
\phantom{1}\\
\phantom{2}
\end{array}
\right\}$}
\end{picture}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{picture}}
\end{picture}
\caption
{\sf Quark-flow diagram for the $\beta$ decay of neutron.}
\label{fig.nucl_particles_quarks_reactions.beta}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Quark confinement}
\label{nucl_particles_quarks_conf}
%------------------------------------------------------
At this point, it is very logical to ask if anybody observed an
isolated quark. The answer is ``no''. Why? And how can one be so
confident of the quark model when no one has ever seen an isolated
quark?\\
Basically, you can't see an isolated quark because the quark-quark
attractive force does not let them go. In contrast to all other
systems, the attraction between quarks grows with the distance
separating them. It is like a rubber cord connecting two balls. When
the balls are close to each other, the cord is not stretched and the
balls do not feel any force. If, however, you try to separate the
balls, the cord pulls them back. The more you stretch the cord, the
stronger the force becomes (according to the Hook's law of
elasticity). Of course, a real rubber cord would eventually
break. This does not happen with the quark-quark force. It can grow to
infinity. This phenomenon is called the {\it confinement of quarks}.\\
Nonetheless, we are sure that the nucleon consists of three quarks
having fractional charges. A hundred years ago Rutherford, by observing
the scattering of charged particles from an atom, proved that its
positive charge is concentrated in a small nucleus. Nowadays, similar
experiments prove the existence of fractional point-like charges inside
the nucleon.\\
The quark model actually is much more complicated than the quark-flow
diagrams. It is a consistent mathematical theory that explains a vast
variety of experimental data. This is why nobody doubts that it
reflects the reality.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Forces of nature}
\label{nucl_particles_forces}
%------------------------------------------------------
If asked how many types of forces exist, many people start counting on
their fingers, and when the count exceeds ten, they answer ``plenty
of''. Indeed, there are gravitational forces , electrical, magnetic,
elastic, frictional forces, and also forces of wind, of expanding
steam, of contracting muscles, etc.\\
If, however, we analyze the root causes of all these forces, we can
reduce their number to just a few fundamental forces (or {\it
fundamental interactions}, as physicists say).\\
For example, the elastic force of a stretched rubber cord is due to
the attraction between the molecules that the rubber is made
of. Looking deeper, we find that the molecules attract each other
because of the electromagnetic attraction between the electrons of one
molecule and nuclei of the other. Similarly, if we depress a piece of
rubber, it resists because the molecules refuse to approach each other
too close due to the electric repulsion of the nuclei. Therefore the
elasticity of rubber has the electromagnetic origin.\\
Any other force in the human world can be analyzed in the same manner.
After doing this, we will find that all forces that we see around us
(in the macroworld), are either of gravitational or electromagnetic
nature. As we also know, in the microworld there are two other types
of forces: The strong (nuclear) forces that act between all hadrons,
and the weak forces that are responsible for changing the quark
flavors.\\
Therefore, all interactions in the Universe are governed by only four
fundamental forces: Strong, electromagnetic, weak and
gravitational. These forces are very different in strength and
range. Their relative strengths are given in Table
\ref{tabl.nucl_particles_forces}. The most strong is the nuclear
interaction. The strength of the electromagnetic forces is one hundred
times lower. The weak forces are nine orders of magnitude weaker than
the nuclear forces, and the gravity is 38 orders of magnitude weaker!
It is amazing that this subtle interaction governs the cosmic
processes. The reason is that the gravitational forces are of long
range and always attractive. There is no such thing as negative mass
that would screen the gravitational field, like negative electrons
screen the field of positive nuclei.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1.3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Force & Relative Strength & Range\\[3mm]
\hline
Strong & 1 & Short\\[3mm]
Electromagnetic & 0.0073 & Long\\[3mm]
Weak & $10^{-9}$ & Very Short\\[3mm]
Gravitational & $10^{-38}$& Long\\[3mm]
\hline
\end{tabular}
\caption{\sf Four fundamental forces and their relative strengths.}
\label{tabl.nucl_particles_forces}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Towards the unified force}
\label{nucl_particles_forces_uni}
%------------------------------------------------------
Physicists always try to simplify things. Since there are only four
fundamental forces, it is tempting to ask "If only four, then why not
only one?". Can it be that all interactions are just different faces
of one master force?\\
The first who started the quest for unification of forces was
Einstein. After completing his general theory of relativity, he spent
30 years in unsuccessful attempts to unify the electromagnetic and
gravity forces. At that time, it seemed logical because both of them
were infinite in range and obeyed the same inverse square law. Einstein
failed because the unification should be done on the basis of quantum
laws, but he tried to do it using the classical concepts.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Electro-weak unification}
\label{nucl_particles_forces_ew}
%------------------------------------------------------
Now it is known that despite the similarities in form of the gravity
and electromagnetic forces, the gravity will be the last to yield to
unification. The more implausible unification of the electromagnetic
and weak forces turned out to be the first successful step towards the
unified interaction.\\
In 1979, the Nobel prize was awarded to Weinberg, Salam, and Glashow,
who developed a unified theory of electromagnetic and weak
interactions. According to that theory, the electromagnetic and weak
forces converge to one {\it electro-weak} interaction at very high
collision energies. The theory also predicted the existence of heavy
particles, the $W$ and $Z$, with masses around 80000\,MeV and
90000\,MeV, respectively. These particles were discovered in 1983,
which brought experimental verification to the new theory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Grand unification}
\label{nucl_particles_forces_grand}
%------------------------------------------------------
The next step was to try to combine the electro-weak theory with the
theory of the strong interactions (i.e. quark theory) in a single
theory. This work was called the {\it grand unification}. Currently,
physicists discuss versions of such theory that predicts the
convergence of the three forces at awfully high energies
$\sim10^{17}$\,MeV. The quarks and leptons in this theory, are the
unified {\it leptoquarks}.\\
The grand unification is not that successful as the electro-weak
theory. It has the problem of mathematical consistency and contradicts
to at least one experiment. The matter is that it predicts the proton
decay,
$$
p\,\longrightarrow\, e^+ +\pi^0\ ,
$$
that does not conserve both the baryon and lepton numbers, with the
lifetime of $\sim10^{29}$\,years. The measurements show, however, that
the lifetime of the proton is at least $10^{32}$\,years.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Theory of everything}
\label{nucl_particles_forces_everything}
%------------------------------------------------------
Some people believe that the grand unification has an inherent
principal flaw. According to them, one cannot unify the forces step by
step (leaving the gravity out), and the correct way is to combine all
four forces in the so-called {\it theory of everything}.\\
There are few different approaches to unifying everything. One of them
suggests that all fundamental particles (quarks and leptons) are just
vibrating modes of string loops in multidimensional space. The
electron is a string vibrating one way, the up-quark is a string
vibrating another way, and so on. The other approach introduces a new
level of fundamental particles, the {\it preons}, that could be
constituent parts of quarks and leptons. The quest goes on.\\
Everyone agrees that constructing the theory of everything would in no
way mean that biology, geology, chemistry, or even physics had been
solved. The universe is so rich and complex that the discovery of the
fundamental theory would not mean the end of science. The ultimate
theory of everything would provide an unshakable pillar of coherence
forever assuring us that the universe is a comprehensible place.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Origin of the universe}
\label{nucl_big_bang}
Looking deep inside microscopic particles, physicists need to collide
them with high kinetic energies. The smaller parts of matter they
want to observe, the higher energy they need. This is why they build
more and more powerful accelerators. However, the accelerators have
natural limitations. Indeed, an accelerator cannot be bigger than the
size of our planet. And even if we manage to build a circular
accelerator around the whole earth (along the equator, for example),
it would not be able to reach the energy of $\sim10^{17}$\,MeV at
which the grand unification of fundamental interactions takes place.\\
So, what are we to do? How can we test the theory of everything? Is it
possible at all? Yes, it is! The astronomically high values, like
$\sim10^{17}$\,MeV, should be looked for in the cosmos, of course. Our
journey towards extremely small objects eventually leads us to
extremely large objects, like whole universe.\\
Equations of Einstein's theory of relativity can describe the
evolution of the universe. Physicists solved these equations back in
time and found that the universe had its beginning. Approximately 15
billion years ago, it started from a zero size point that exploded and
rapidly expanded to the present tremendous scale. At the first
instants after the explosion, the matter was at such incredibly high
density and temperature that all particles had kinetic energies even
higher than the unification energy $\sim10^{17}$\,MeV. This means that
at the very beginning there was only one sigle force and no difference
among fundamental particles. Everything was unified and ``simple''.\\
You may ask ``So what? How can so distant past help us?''. In many
ways! The development of the universe was governed by the fundamental
forces. If our theories about them are correct, we should be able to
reproduce (with calculations) how that development proceeded step by
step. During the expansion, all the nuclei and atoms in the cosmos
were created. The amounts of different nuclei are not the same. Why?
Their relative abundances were determined by the processes in the
first moments after the explosion. Thus, comparing what follows from
the theories with the observed abundances of chemical elements, we can
judge validity of our theories.\\
Nowadays, the most popular theory, describing the history of the
universe, is the so--called {\it Big-Bang} model. The
diagram given in Fig.~\ref{bigbang.fig}, shows the sequence of events
which led to the creation of matter in its present form.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure bigbang.fig %%%%%%%%%%%%%%%%%%%%
\begin{figure}[h!tb]
\begin{center}
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\begin{picture}(80,57)
%%%%%%%%% Frame %%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\put(13,-6.5){\begin{picture}(0,0)%
\put(15,57){\Large\sf BIG\quad BANG}
\put(10,59){$\star$}
\put(42,58){$\star$}
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\put(30,61){$\star$}
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%
\put(22,14){\sf today}
\put(51,7){\sf time}
\put(-11,7){\sf temperature}
\put(51,54.5){\small 0}
\put(51,51,2){\small $10^{-43}$sec}
\put(51,48){\small $10^{-35}$sec}
\put(51,44.4){\small $10^{-10}$sec}
%\put(51,40.8){\small $10^{-4}$sec}
\put(51,35.4){\small 1\,sec}
\put(51,29){\small 10\,sec}
\put(51,21){\small 500\,sec}
\put(51,14.3){\small $15\times10^9$years}
\put(0,14.3){\llap{\small $2.9$\,K}}
\put(0,21){\llap{\small $10^7$\,K}}
\put(0,29){\llap{\small $10^{9}$\,K}}
\put(0,35.4){\llap{\small $10^{10}$\,K}}
%\put(0,40.8){\llap{\small $10^{12}$\,K}}
\put(0,44.4){\llap{\small $10^{15}$\,K}}
\put(0,48){\llap{\small $10^{28}$\,K}}
\put(0,51.2){\llap{\small $10^{32}$\,K}}
\put(16,52.5){\small single unified force}
\put(10,49.3){\small gravitational force separated}
%\put(20,46.3){\small lepto-quarks,\quad $X$}
\put(14,45.7){\small strong force separated}
%\put(12,42.7){\small hadrons, leptons,\quad $Z, W$}
\put(15,42.1){\small weak force separated}
\put(8,38){\small
$n\!+\!\nu\!\to\! p\!+\!e^-$,\qquad $p\!+\!{\bar\nu}\!\to\! n\!+\!e^+$}
\put(8,32){\small
$p\!+\!n\!\to\! {}^2{\rm H}\!+\!\gamma$,\quad ${}^2{\rm H}\!+
\!{}^2{\rm H}\!\to\!{}^4{\rm He}\!+\!\gamma$}
\put(20,25){\small $pp$--chain}
\end{picture}}
\end{picture}
\end{center}
\caption{\sf Schematic ``history'' of the universe.}
\label{bigbang.fig}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Nobody knows what was before the Big Bang and why it happened, but it
is assumed that just after this enigmatic cataclysm, the universe was
so dense and hot that all four forces of nature (strong,
electromagnetic, weak, and gravitational) were indistinguishable and
therefore gravity was governed by quantum laws, like the other three
types of interactions. A complete theory of quantum gravity has not
been constructed yet, and this very first ``epoch'' of our history
remains as enigmatic as the Big Bang itself.\\
The ideal ``democracy'' (equality) among the forces lasted only a
small fraction of a second. By the time $t\sim 10^{-43}$\,sec the
universe cooled down to $\sim 10^{32}$\,K and the gravity
separated. The other three forces, however, remained unified into one
universal interaction mediated by an extremely heavy particle, the
so-called $X$ boson, which could transform leptons into quarks and
vice versa.\\
When at $t\sim 10^{-35}$\,sec most of the $X$ bosons decayed, the
quarks combined in trios and pairs to form nucleons, mesons, and other
hadrons. The only symmetry which lasted up to $\sim10^{-10}$\,sec, was
between the electromagnetic and weak forces mediated by the $Z$ and
$W$ particles. From the moment when this last symmetry was broken
($\sim 10^{-10}$\,sec) until the universe was about one second old,
neutrinos played the most significant role by mediating the
neutron-proton transmutations and therefore fixing their balance
(neutron to proton ratio).\\
Already in a few seconds after the Big Bang nuclear reactions started to
occur. The protons and neutrons combined very rapidly to form
deuterium and then helium. During the very first seconds there were
too many very energetic photons around which destroyed these nuclei
immediately after their formation. Very soon, however, the continuing
expansion of the universe changed the conditions in favour of these
newly born nuclei. The density decreased and the photons could not
destroy them that fast anymore.\\
During a short period of cosmic history, between about 10 and 500
seconds, the entire universe behaved as a giant nuclear fusion reactor
burning hydrogen. This burning took place via a chain of nuclear
reactions, which is called the {\it $pp$-chain} because the first
reaction in this sequence is the proton-proton collision leading to
the formation of a deuteron. Nowadays, the same $pp$-chain is the main
source of energy in our sun and other stars.\\
But how do we know that the scenario was like this? In other words,
how can we check the Big--Bang theory? Is it possible to prove
something which happened 15 billion years ago and in such a short
time? Yes, it is! The $pp$-chain fusion,\\[5mm]
%%%%%%
\raisebox{0.5cm}{\parbox{4cm}{\qquad\underline{\it pp-chain:}}}
\parbox{5cm}{\fbox{$
\begin{array}{rcl}
{\rm p+p} \!\!&\to&\!\! {}^2{\rm H+e}^+ +\nu_e \\
%
{\rm e}^-{\rm +p+p} \!\!&\to&\!\! {}^2{\rm H}+\nu_e \\
%
{\rm p}+{}^2{\rm H} \!\!&\to&\!\! {}^3{\rm He} +\gamma \\
%
{}^3{\rm He} +{}^3{\rm He} \!\!&\to&\!\! {}^4{\rm He + p + p}\\
%
{}^3{\rm He}+{}^4{\rm He} \!\!&\to&\!\!{}^7{\rm Be}+\gamma \\
\end{array}
$}}
\begin{center}
{\huge$\swarrow\quad\searrow $}\\[0.2cm]
\parbox{0.7cm}{\phantom{-}}
\fbox{\parbox{4cm}{$\!\!
\begin{array}{rcl}
{\rm e^-\!\!+\!{}^7Be} \!\!\!\!&\to&\!\!\!\! {}^7{\rm Li} +\nu_e\\
{\rm p}+{}^7{\rm Li} \!\!\!\!&\to&\!\!\!\! {}^8{\rm Be}+\gamma\\
{}^8{\rm Be} \!\!\!\!&\to&\!\!\!\! {}^4{\rm He} + {}^4{\rm He}\\
\end{array}
$}}\ \
\fbox{\parbox{4.5cm}{$\!\!
\begin{array}{rcl}
{\rm p}+{}^7{\rm Be} \!\!\!\!&\to&\!\!\!\! {}^8{\rm B} +\gamma \\
{}^8{\rm B}\!\!\!\!&\to&\!\!\!\!
{}^8{\rm Be}{}^* +{\rm e}^+\! +\!\nu_e\\
{}^8{\rm Be}^* \!\!\!\!&\to&\!\!\!\! {}^4{\rm He} + {}^4{\rm He}\\
\end{array}
$}}
\end{center}
is the key for such a proof.\\
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure DHe.fig %%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht!b]
\begin{center}
\unitlength=1mm
\begin{picture}(90,40)
%
\put(10,3){\begin{picture}(0,0)%
\put(0,0){\vector(1,0){80}}
\put(0,0){\vector(0,1){35}}
\multiput(0,0)(20,0){4}{\line(0,-1){2}}
\multiput(0,0)(0,5){7}{\line(-1,0){2}}
%
\put(-4,-1.5){\small\llap{$10^{-12}$}}
\put(-4,3.5){\small\llap{$10^{-10}$}}
\put(-4,8.5){\small\llap{$10^{-8}$}}
\put(-4,13.5){\small\llap{$10^{-6}$}}
\put(-4,18.5){\small\llap{$10^{-4}$}}
\put(-4,23.5){\small\llap{$10^{-2}$}}
\put(-4,28.5){\small\llap{$1$}}
\put(5,38){\llap{$\rho/\rho_p$}}
%
\put(-2,-6){\small$10$}
\put(18,-6){\small$10^2$}
\put(38,-6){\small$10^3$}
\put(58,-6){\small$10^4$}
\put(80,-6){\llap{$t$ (sec)}}
%
% ---- deuterium ------
\put(0,2.899){\special{em:moveto}}
\put(3.509,4.167){\special{em:lineto}}
\put(7.018,5.616){\special{em:lineto}}
\put(10.526,7.065){\special{em:lineto}}
\put(14.035,9.601){\special{em:lineto}}
\put(17.544,11.957){\special{em:lineto}}
\put(21.053,14.855){\special{em:lineto}}
\put(24.561,18.841){\special{em:lineto}}
\put(28.070,23.551){\special{em:lineto}}
\put(28.772,24.094){\special{em:lineto}}
\put(29.474,24.638){\special{em:lineto}}
\put(29.825,24.819){\special{em:lineto}}
\put(30.175,24.638){\special{em:lineto}}
\put(30.877,24.094){\special{em:lineto}}
\put(31.579,23.551){\special{em:lineto}}
\put(35.088,21.377){\special{em:lineto}}
\put(38.596,20.652){\special{em:lineto}}
\put(42.105,20.290){\special{em:lineto}}
\put(45.614,20.109){\special{em:lineto}}
\put(75,20.109){\special{em:lineto}}
\put(70,16){\llap{\sl deuterium}}
% ---- helium ------
\put(5.263,0){\special{em:moveto}}
\put(7.018,2.174){\special{em:lineto}}
\put(10.526,7.33){\special{em:lineto}}
\put(14.035,10.3){\special{em:lineto}}
\put(17.544,11.775){\special{em:lineto}}
\put(21.053,13.949){\special{em:lineto}}
\put(24.561,17){\special{em:lineto}}
\put(28.070,24.819){\special{em:lineto}}
\put(29.235,27.174){\special{em:lineto}}
\put(30.407,28.261){\special{em:lineto}}
\put(31.579,28.442){\special{em:lineto}}
\put(35.088,28.623){\special{em:lineto}}
\put(75,28.623){\special{em:lineto}}
\put(67,24.5){\llap{\sl helium}}
%--------
\end{picture}}
\end{picture}
\end{center}
\caption{\sf Mass fractions $\rho$ (relative to hydrogen $\rho_p$)
of primordial deuterium and
$^4$He versus the time elapsed since the Big Bang.}
\label{DHe.fig}
\end{figure}
%%%%%%%%%%%%%%%
As soon as the nucleosynthesis started, the amount of deuterons,
helium isotopes, and other light nuclei started to increase. This is
shown in Fig.~\ref{DHe.fig} for $^2$H and $^4$He. The temperature and
the density, however, continued to decrease. After a few minutes the
temperature dropped to such a level that the fusion practically
stopped because the kinetic energy of the nuclei was not sufficient to
overcome the electric repulsion between nuclei anymore. Therefore the
abundances of light elements in the cosmos were fixed (we call them
the {\it primordial abundances}). Since then, they practically remain
unchanged, like a photograph of the past events, and astronomers can
measure them. Comparing the measurements with the predictions of the
theory, we can check whether our assumptions about the first seconds
of the universe are correct or not.\\
Astronomy and the physics of microworld come to the same point from
different directions. The Big Bang theory is only one example of their
common interest. Another example is related to the mass of
neutrino. When Pauli suggested this tiny particle to explain the
nuclear $\beta$-decay, it was considered as massless, like the
photon. However, the experiments conducted recently, indicate that
neutrinos may have small non-zero masses of just a few eV.\\
In the world of elementary particles, this is extremely small mass,
but it makes a huge difference in the cosmos. The universe continues
to expand despite the fact that the gravitational forces pull
everything back to each other. The estimates show, that the visible
mass of all galaxies is not sufficient to stop and reverse the
expansion. The universe is filled with a tremendous number of
neutrinos. Even with few eV per neutrino, this amounts to a huge total
mass of them, which is invisible but could reverse the expansion.\\
Thus, the cooperation of astronomers and particle physicists has led
to significant advances in our understanding of the universe and its
evolution. The quest goes on. A famous German philosopher Friedrich
Nietzsche once said that ``The most incomprehensible thing about this
Universe is that it is comprehensible.''
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