NUCLEI

• Atomic Mass:

Atomic mass in an atom or group of an atom is the sum of the masses of protons, neutrons and electrons. For eg. the mass of a carbon atom,${}^{12}$C , is $1.992647\times 10^{-26}kg$ and the atomic mass unit (u), is defined as $\frac{1}{12}$th of the mass of the carbon (${}^{12}$C) atom.
$1u=\frac{1.992647\times 10^{-26}}{12}=1.660539\times 10^{-27}kg.$
Accurate measurement of atomic masses is carried out with a mass spectrometer.

• Composition of Nucleus:

The composition of a nucleus can be described using the following terms and symbols meh:
$Z$ - atomic number = number of protons
$N$ - neutron number = number of neutrons
$A$ - mass number = $Z+N$ = total number of protons and neutrons
Protons or Neutrons are also called Nucleons. Hence, the mass number ($A$) of an atom is the total number of nucleons in it. A typical nuclide of an atom is symbolized with the notation,${}_Z^{A}X$, where $X$ is the chemical symbol of the atom.

Since the nucleus is made up of positively charged protons and electrically neutral neutrons, the overall charge of the nucleus is positive and it has the value of $+Ze$. But the atom is electrically neutral which implies that the number of electrons in the atom is equal to the number of protons in the nucleus.

Isotopes:
isotopes are atoms of the same element having the same atomic number $Z$, but different mass number $A$.
Eg. Isotopes of Hydrogen, ${}_1^{1}H$(Hydrogen), ${}_1^{2}H$(Deuterium) , ${}_1^{3}H$ (Tritium)

Isobars:
Isobars are the atoms of different elements having the same mass number $A$, but different atomic number $Z$.
Example: ${}_{19}^{40}K$ and ${}_{20}^{40}C$

Isotones:
Isotones are the atoms of different elements having the same number of neutrons.
Example: ${}_5^{12}Ba$ and ${}_6^{13}C$

Size of the Nucleus:
It has been found that a nucleus of mass number A has a radius
$R=R_o{A}^{\frac{1}{3}}$
where $R_o=1.2\times 10^{-15}m(=1.2fm)$. This means the volume of the nucleus, which is proportional to $R^3$ is proportional to $A$.

Density of Nucleus:
$\rho =\frac{Massofnucleus}{VolumeofNucleus}=\frac{Am}{\frac{4}{3}\pi R_o^{3}A}=2.7\times 10^{17}kg/m^3$
Thus the density of the nucleus is constant, independent of $A$, for all nuclei. Different nuclei are like a drop of liquid of constant density.

Mass Energy:
Einstein gave the famous mass-energy equivalence relation
$E=m{c}^2$
Where m is the mass
$c$ is the velocity of light in vacuum and is approximately equal to $3\times 10^{8}m{s}^{-1}$
According to this relation, the mass can be converted into energy and energy can be converted into mass.

Nuclear Binding Energy:

Mass Defect
The nucleus is made up of neutrons and protons. Therefore, it may be expected that the mass of the nucleus is equal to the total mass of its individual protons and neutrons. However, the nuclear mass $M$ is found to be always less than this.
The difference in mass of a nucleus and its constituents,$\Delta M$, is called mass defect and is given by, $\Delta m=\left[Z{m}_p+(A-Z)m_n-M\right]$

Binding Energy
If a certain number of neutrons and protons are brought together to form a nucleus of a certain charge and mass, an energy $E_b$ will be released in the process. The energy $E_b$ is called the binding energy of the nucleus. If we separate a nucleus into its nucleons, we would have to supply a total energy equal to $E_b$, to those particles.
Using the mass-energy equivalence we can write,
$E_b=\Delta m{c}^2=\left[Z{m}_p+(A-Z)m_n-M\right]c^2$
Thus higher the binding energy, more stable is the nucleus.
The binding energy per nucleon, $E_{bn}$, is the ratio of the binding energy $E_b$ of a nucleus to the number of the nucleons, $A$, in that nucleus: $E_{bn}=E_b/A$
We can think of binding energy per nucleon as the average energy per nucleon needed to separate a nucleus into its individual nucleons

The binding energy curve

Here are the conclusions that we can draw from the binding energy curve:

• The value of BE rises as the mass number increases until it reaches a maximum value of $8.8MeV$ for $A=56(iron)$ and then it slowly decreases.
• The binding energy per nucleon, $E_{bn}$ , is practically constant, i.e., practically independent of the atomic number for nuclei of middle mass number $(30<A<170)$.
• $E_{bn}$ is lower for both light nuclei $(A<30)$and heavy nuclei $(A>170)$.

Nuclear Force:

 

1. The nuclear force is much stronger than the Coulomb force acting between charges or the gravitational forces between masses. The nuclear binding force has to dominate over the Coulomb repulsive force between protons inside the nucleus.
2. The nuclear force is short ranged and the force between two nucleons falls rapidly to zero as their distance is more than a few femtometers. This leads to saturation of forces in a medium or a large-sized nucleus, which is the reason for the constancy of the binding energy per nucleon.

The potential energy between two nucleons is a minimum at a distance $r_o$ of about $0.8fm$. This means that the force is attractive for distances larger than $0.8fm$ and repulsive if they are separated by distance more than $0.8fm$

3. The nuclear force between neutron-neutron, proton-neutron and proton-proton is approximately the same. The nuclear force does not depend on the electric charge.

Activity
The total decay rate R of a sample is the number of nuclei disintegrating per unit time also called the activity of the sample.
$R=-\frac{dN}{dt}=\lambda N_0{e}^{-\lambda t}=R_o{e}^{-\lambda t}$
SI unit: $1becquerel$ is simply equal to $1$ disintegration or decay per second.
  Half-lifeof a radionuclide (denoted by $T_{\frac{1}{2}}$) is the time it takes for a sample that has initially, say $N_0$ radio nuclei to reduce to $N_0/2$.
$T_{\frac{1}{2}}=\frac{ln\frac{1}{2}}{\lambda }=\frac{0.693}{\lambda }$

Mean Lifetime:
The mean lifetime of the nucleus is the ratio of sum or integration of life timesof all nuclei to the total number nuclei present initially.
$\tau =\frac{\text{Total lifetime of}{N}_o\text{nuclei}}{N_o}=\frac{{\int }_o^{\infty }t[dN]}{N_o}=\frac{1}{\lambda }$
 

• Decay Pathways:

Radioactive elements can decay into stable elements undergoing various schemes, namely,alpha decay, beta decayorgamma decay. Let us tabulate the differences between different kind of radioactive emissions:

Alpha Decay:
In this process an unstable nucleus emits a helium nucleus,${}^{4}H{e}_2$, which contains two protons and two neutrons. Hence, the mass number and the atomic number of the daughter nucleus decreases by four and two, respectively. Thus, the transformation of a nucleus${}^A{X}_Z$ into a nucleus${}^{A-4}{Y}_{Z-2}$ can be expressed as,
${}^A{X}_Z{\to }^{A-4}{Y}_{Z-2}{+}^{4}H{e}_2$
where${}^A{X}_Z$is the parent nucleus and${}^{A-4}{Y}_{Z-2}$is the daughter nucleus.
For example, when ${}^{238}{U}_{92}$ undergoes alpha-decay, it transforms to ${}^{234}{U}_{90}$
${}^{238}{U}_{92}{\to }^{234}{U}_{90}{+}^{4}H{e}_2$
Beta Decay:
A nucleus that decays spontaneously by emitting an electron or a positron is said to undergo beta decay. In beta minus (${\beta }^{-}$) decay, an electron is emitted by the nucleus and a neutron transforms into a proton within the nucleus according to$n\to p+e^{-}+\bar{v}$
In beta plus (${\beta }^{+}$) decay, a positron is emitted by the nucleus and a proton transforms into neutron (inside the nucleus) via $p\to n+e^{+}+v$

Gamma Decay:
Like an atom, a nucleus also has discrete energy levels - the ground state and excited state. The scale of energy is, however, at the range of MeV. When a nucleus in an excited state, spontaneously decays to its ground state (or to a lower energy state), a photon is emitted with energy equal to the difference in the two energy levels of the nucleus. This is the so-called gamma decay. The energy (MeV) corresponds to radiation of extremely short wavelength, shorter than the hard X-ray region. Typically, a gamma ray is emitted when a $\alpha$ or $\beta$ decay results in a daughter nucleus in an excited state.
${}_Z^A{X}^{\ast }{\to }_Z^{A}X+\text{gamma}(\gamma )\text{rays}$
Here Asterix(*) means excited state

• Discovery of Neutrons:

In 1932 James Chadwick discovered that when Beryllium is bombarded with alpha particles, the kind of radiation that is emitted is not gamma radiation but a stream of neutral particles which are called neutrons. The reaction is: ${}_4^{9}Be{+}_2^{4}He{\to }_6^{12}C{+}_0^{1}n$
Neutrons are stable inside the nucleus but outside the nucleus they are unstable and decay fast with a mean lifetime of 15 min. If the neutron comes out of the nucleus (free neutron), it decays with the emission of a beta particle. The neutron decay equation is
$n\to p+e^{-}+\overline{\nu }$

• Nuclear Fission:

The process of breaking up of the nucleus of a heavier atom into two smaller nuclei with the release of a large amount of energy is called nuclear fission.
A very heavy nucleus, say $A=240$, has lower binding energy per nucleon compared to that of a nucleus with $A=120$. Thus if a nucleus $A=240$ breaks into two $A=120$ nuclei, nucleons get more tightly bound. This implies energy would be released in the process.
Example:
${}^1{n}_0{+}^{235}{U}_{92}{\to }^{236}{U}_{92}{\to }^{133}S{b}_{51}{+}^{99}N{b}_{41}+4^1{n}_0$

Chain reaction:
${}_{92}^{235}U+{}_0^{1}n\to {{}_{92}^{236}U}^{\ast }\to {}_{56}^{141}Ba+{}_{36}^{92}Kr+3{}_0^{1}n+Q$
When one $U-235$ nucleus undergoes fission, about $200MeV$ of energy is released. Butfrom each fission reaction, three neutrons are released. The average energy of a neutron produced in fission of $U-235$ is $2MeV$ . These three neutrons cause further fission in another three $U-235$ nuclei which in turn produce nine neutrons. These nine neutrons initiate fission in another $27U-235$ nuclei and so on. This is called a chain reaction

• Nuclear Fusion:

Nuclear fusion is a process when two or more nuclei combine to form a large nucleus and release some energy. Consider two very light nuclei ($A\le 10$) joining to form a heavier nucleus. The binding energy per nucleon of the fused heavier nuclei is more than the binding energy per nucleon of the lighter nuclei. This means that the final system is more tightly bound than the initial system so energy would be released.
The examples of Nuclear Fusion Reaction is:
${}^6{C}_{12}{+}^1{H}_1{\to }^7{C}_{14}+4.3MeV$

• Nuclear Reactor:

Nuclear reactor is a system in which the nuclear fission takes place in a self-sustained and controlled manner.

A nuclear reactor is made of fuel, moderator, control rods, coolant, pressure vessel or pressure tubes, steam generator and containment. Let us understand its working in detail.
Fuel: The fuel is fissionable material, usually uranium or plutonium. In addition to this, a neutron source is required to initiate the chain reaction for the first time.
Moderators: The moderator is a material used to convert fast neutrons into slowneutrons. Usually, the moderators are chosen in such a way that it must be very light nucleushaving mass comparable to that of neutrons. Most of the reactors use water, heavy water ($D_{2}O$) and graphite as moderators.
Control rods: The control rods are used to adjust the reaction rate. During each fission,on an average $2.5neutrons$ are emitted and in order to have the controlled chain reactions, only one neutron is allowed to cause another fission and the remaining neutrons are absorbed by the control rods. Usually cadmium or boron acts as control rod material.
Multiplication Factor ($K$) is the ratio of number of fission produced by a given generation of neutrons to the number of fission of the preceding generation. It is the measure of the average number of neutrons produced per fission. For $K=1$, the operation of the reactor is said to be critical, which is required for steady power operation.
Cooling system: The cooling system removes the heat generated in the reactor core. Ordinary water, heavy water and liquid sodium are used as coolant since they have very high specific heat capacity.