Introduction

Building upon the understanding from the previous chapter (Atoms), we know that an atom consists of a central, dense nucleus containing positive charge and most of the atom's mass, surrounded by orbiting electrons. The $\alpha$-particle scattering experiments revealed that the nucleus is significantly smaller than the atom, with its radius being about $10^{-4}$ times that of the atomic radius. This implies that the volume of the nucleus is approximately $(10^{-4})^3 = 10^{-12}$ times the volume of the atom, meaning atoms are largely empty space. Despite its tiny size, the nucleus holds over 99.9% of the atom's mass.

This chapter delves into the nucleus itself. Does the nucleus have an internal structure? What are the particles that make up the nucleus, and how are they held together within such a small volume? We will explore the properties of nuclei, including their size, mass, stability, and the fascinating phenomena associated with nuclear transformations, such as radioactivity, nuclear fission, and nuclear fusion.

Atomic Masses And Composition Of Nucleus

Atomic masses are extremely small when measured in kilograms. To work with these small values conveniently, a special unit called the atomic mass unit (u) is used. The atomic mass unit is defined as one-twelfth (1/12th) the mass of a carbon-12 ($^{12}\text{C}$) atom. Experimentally, it is found that:

$$1 \text{ u} = \frac{\text{Mass of one }^{12}\text{C atom}}{12} \approx 1.660539 \times 10^{-27} \text{ kg}$$

Atomic masses of elements, when expressed in atomic mass units, are often close to integer multiples of the mass of a hydrogen atom ($^1\text{H}$), but there are exceptions, like chlorine (average atomic mass $\approx 35.46$ u).

Accurate measurements of atomic masses using instruments like mass spectrometers revealed the existence of isotopes. Isotopes are atoms of the same element that have the same chemical properties (due to the same number of electrons and protons) but differ in mass. Most elements are found to be a mixture of several isotopes with varying relative abundances. For example, chlorine exists as two main isotopes, approximately $^{35}\text{Cl}$ (mass $\approx 34.98$ u) and $^{37}\text{Cl}$ (mass $\approx 36.98$ u), with abundances of about 75.4% and 24.6%, respectively. The weighted average of these isotopic masses gives the observed atomic mass of chlorine.

Even hydrogen has three isotopes: $^{1}\text{H}$ (protium), $^{2}\text{H}$ (deuterium), and $^{3}\text{H}$ (tritium). The nucleus of the most common hydrogen isotope ($^1\text{H}$) is called a proton. Its mass is $m_p \approx 1.00727$ u. This mass is approximately the mass of a hydrogen atom minus the mass of an electron.

We know that the number of electrons in a neutral atom is equal to its atomic number $Z$. Since the atom is electrically neutral, the positive charge must be located in the nucleus and equal to $+Ze$. As each proton carries a charge of $+e$, the number of protons in the nucleus of an atom is exactly equal to its atomic number $Z$.

The masses of the isotopes of hydrogen ($^1\text{H}, ^2\text{H}, ^3\text{H}$) are approximately in the ratio 1:2:3. Since they all have only one proton, their nuclei must contain additional neutral matter to account for the extra mass in deuterium and tritium. This led to the hypothesis that nuclei contain neutral particles in addition to protons.

The existence of these neutral particles, called neutrons, was experimentally confirmed by James Chadwick in 1932. He observed that bombarding beryllium with $\alpha$-particles produced neutral radiation that could knock protons out of other nuclei. By applying conservation laws of energy and momentum, Chadwick determined that this radiation consisted of particles with mass very close to that of a proton. The mass of a neutron is $m_n \approx 1.00866$ u. Chadwick received the Nobel Prize in Physics in 1935 for this discovery.

Unlike free protons, free neutrons are unstable and decay into a proton, an electron, and an antineutrino with a mean lifetime of about 1000 seconds. However, neutrons are stable when bound within a nucleus.

The composition of a nucleus can now be described by:

• Atomic Number (Z): The number of protons in the nucleus. This determines the element's identity.
• Neutron Number (N): The number of neutrons in the nucleus.
• Mass Number (A): The total number of protons and neutrons in the nucleus. $A = Z + N$.

Protons and neutrons together are called nucleons. The mass number A is the total number of nucleons in the nucleus. A specific nuclear species or nuclide is denoted by $\text{X}^A_Z$, where X is the chemical symbol.

Using this notation, isotopes of an element X have the same Z but different N (and hence different A). For example, $^{1}\text{H}_1$, $^{2}\text{H}_1$, and $^{3}\text{H}_1$ are isotopes of hydrogen (Protium, Deuterium, Tritium). Gold ($_{79}\text{Au}$) has 79 protons, and its isotope $^{197}_{79}\text{Au}$ has $197 - 79 = 118$ neutrons.

Nuclides with the same mass number A are called isobars (e.g., $^{3}_{1}\text{H}$ and $^{3}_{2}\text{He}$). Nuclides with the same neutron number N but different atomic number Z are called isotones (e.g., $^{198}_{80}\text{Hg}$ ($N=118$) and $^{197}_{79}\text{Au}$ ($N=118$)).

Size Of The Nucleus

Early estimates of nuclear size came from Rutherford's scattering experiments. The distance of closest approach of an $\alpha$-particle in a head-on collision with a nucleus provided an upper limit to the nuclear radius. For a 5.5 MeV $\alpha$-particle scattering off a gold nucleus, the distance of closest approach was about $4.0 \times 10^{-14}$ m. This suggested the gold nucleus was smaller than this value.

More precise measurements of nuclear sizes have been made using scattering experiments with high-energy electrons, which interact primarily with the nucleus's charge distribution. These experiments show that the radius ($R$) of a nucleus is approximately related to its mass number ($A$) by the formula:

$$R = R_0 A^{1/3}$$

where $R_0$ is an empirical constant, approximately $1.2 \times 10^{-15}$ m or 1.2 fm (fermi). Since the volume of a sphere is proportional to $R^3$, the nuclear volume is proportional to $(A^{1/3})^3 = A$.

The density of nuclear matter is given by Mass/Volume. The mass of a nucleus is approximately $A \times u$ (where $u$ is the atomic mass unit). The volume is approximately $\frac{4}{3}\pi R^3 = \frac{4}{3}\pi (R_0 A^{1/3})^3 = \frac{4}{3}\pi R_0^3 A$.

Nuclear density $\rho_{nuclear} \approx \frac{A \times u}{\frac{4}{3}\pi R_0^3 A} = \frac{u}{\frac{4}{3}\pi R_0^3}$.

Since $u$, $\pi$, and $R_0$ are constants, the nuclear matter density is approximately constant for all nuclei, regardless of their mass number A. Calculating this density gives a value around $2.3 \times 10^{17} \text{ kg/m}^3$. This is extremely high compared to the density of ordinary matter (e.g., water $\approx 10^3 \text{ kg/m}^3$), highlighting how densely packed the mass is within the nucleus. This supports the idea that most of the atom's volume is empty space.

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Mass-Energy And Nuclear Binding Energy

Mass – Energy

Einstein's theory of special relativity established the fundamental equivalence between mass and energy. This means that mass is a form of energy and can be converted into other forms of energy, and vice versa. The famous mass-energy equivalence relation is:

$$E = mc^2$$

where $E$ is the energy equivalent of mass $m$, and $c$ is the speed of light in vacuum (approximately $3 \times 10^8 \text{ m/s}$). This principle implies that the separate conservation laws of mass and energy, which were previously thought to be independent, are actually part of a single, unified law: the conservation of mass-energy.

The energy equivalent of even a small amount of mass is enormous due to the large value of $c^2$. This conversion of mass into energy is most evident and significant in nuclear reactions, where energy releases are orders of magnitude larger than in chemical reactions.

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Nuclear Binding Energy

Nuclei are composed of protons and neutrons (nucleons). A fundamental observation is that the total mass of a stable nucleus ($M$) is always less than the sum of the individual masses of the protons and neutrons that constitute it in a free, unbound state. This difference in mass is called the mass defect ($\Delta M$).

For a nucleus with $Z$ protons and $N$ neutrons (mass number $A = Z+N$), the expected total mass of the constituents is $Z m_p + N m_n$, where $m_p$ is the mass of a free proton and $m_n$ is the mass of a free neutron. The mass defect is defined as:

$$\Delta M = [Z m_p + (A - Z) m_n] - M$$

Note: For atomic mass, the mass defect is sometimes calculated using atomic masses, where $[Z m(^1\text{H}) + (A-Z)m_n] - M_{atom}$. Since $m(^1\text{H}) = m_p + m_e$, this is equivalent to $[Z m_p + Z m_e + (A-Z)m_n] - (M_{nucleus} + Z m_e) = [Z m_p + (A-Z)m_n] - M_{nucleus}$. So the definition using nuclear mass is standard.

According to Einstein's mass-energy equivalence, this mass defect corresponds to an equivalent amount of energy, given by:

$$E_b = \Delta M c^2$$

This energy $E_b$ is called the binding energy of the nucleus. It represents the energy that must be supplied to the nucleus to break it apart into its constituent protons and neutrons. Conversely, it is the energy released when the individual nucleons combine to form the nucleus. A larger binding energy indicates a more stable nucleus.

Often, a more useful quantity for comparing the stability of different nuclei is the binding energy per nucleon ($E_{bn}$). This is the binding energy divided by the total number of nucleons (mass number A):

$$E_{bn} = \frac{E_b}{A}$$

The binding energy per nucleon represents the average energy required to remove a single nucleon from the nucleus. A plot of the binding energy per nucleon ($E_{bn}$) versus the mass number ($A$) provides valuable insights into nuclear stability (

Figure 13.1):

• The curve rises rapidly for light nuclei ($A < 30$).
• It reaches a broad peak for intermediate mass nuclei ($30 < A < 170$), where $E_{bn}$ is relatively constant, around 8 MeV. Iron-56 ($^{56}\text{Fe}$, $A=56$) has the maximum binding energy per nucleon ($\approx 8.75$ MeV), making it one of the most stable nuclei.
• The curve gradually decreases for heavy nuclei ($A > 170$).

This shape of the binding energy curve has significant implications for nuclear reactions and energy generation, suggesting that energy can be released through processes where less tightly bound nuclei transform into more tightly bound nuclei.

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Nuclear Force

Within the tiny volume of the nucleus, positively charged protons are packed together. The electrostatic Coulomb force between these protons is strongly repulsive. The fact that the nucleus remains bound together despite this repulsion indicates the existence of a powerful attractive force acting between the nucleons (protons and neutrons). This force is called the strong nuclear force or simply the nuclear force.

Key features of the nuclear force:

• It is the strongest fundamental force in nature. It is significantly stronger than the electromagnetic (Coulomb) force and much, much stronger than the gravitational force at nuclear distances. Its strength is what overcomes the Coulomb repulsion between protons and binds both protons and neutrons together.
• It is a short-range force. The nuclear force is attractive at distances typical within the nucleus (around 0.8 fm to a few femtometers), but it falls off very rapidly and becomes negligible at distances greater than a few femtometers.
• The potential energy between two nucleons as a function of their separation shows a minimum at a distance $r_0 \approx 0.8$ fm (
  Figure 13.2). This minimum indicates an attractive force for $r > r_0$. At very short distances ($r < r_0$), the force becomes strongly repulsive, preventing nucleons from collapsing into each other.
• The short-range nature of the nuclear force explains the approximate constancy of the binding energy per nucleon for intermediate and heavy nuclei (saturation property). A nucleon inside a large nucleus only interacts with its nearest neighbors within the range of the force, rather than with all other nucleons. Adding more nucleons to a large nucleus doesn't significantly increase the binding energy per nucleon of the interior nucleons.
• The nuclear force is essentially charge-independent. The attractive force between two protons (p-p), a proton and a neutron (p-n), and two neutrons (n-n) is approximately the same, apart from the additional Coulomb repulsion between protons.

Unlike the Coulomb force or gravity, there is no simple mathematical formula to precisely describe the nuclear force. Its complex nature arises from the interactions between the quarks and gluons within the nucleons.

Radioactivity

Radioactivity is the spontaneous disintegration or decay of unstable atomic nuclei, accompanied by the emission of radiation. This phenomenon was accidentally discovered by A.H. Becquerel in 1896 while studying luminescence from uranium compounds. He found that a uranium salt could expose a photographic plate even when wrapped in opaque paper, indicating the emission of penetrating radiation.

Later experiments by Marie Curie, Pierre Curie, and others revealed that radioactivity is a nuclear process and that there are different types of emissions. The three main types of radioactive decay found in nature are:

• Alpha ($\alpha$) decay: An unstable nucleus emits an alpha particle, which is a helium nucleus ($^4_2\text{He}$, consisting of 2 protons and 2 neutrons). The parent nucleus transforms into a daughter nucleus with a mass number reduced by 4 and an atomic number reduced by 2.
• Beta ($\beta$) decay: This involves the emission of a beta particle, which can be either an electron ($\beta^-$) or a positron ($\beta^+$). In $\beta^-$ decay, a neutron in the nucleus is converted into a proton, an electron, and an antineutrino. The daughter nucleus has the same mass number but an atomic number increased by 1. In $\beta^+$ decay, a proton is converted into a neutron, a positron, and a neutrino. The daughter nucleus has the same mass number but an atomic number decreased by 1.
• Gamma ($\gamma$) decay: After undergoing $\alpha$ or $\beta$ decay, the daughter nucleus is often left in an excited energy state. It transitions to a lower energy state (usually the ground state) by emitting a gamma ray, which is a high-energy photon (electromagnetic radiation). Gamma decay does not change the mass number or atomic number of the nucleus but reduces its internal energy.

Radioactive decay is a statistical process; while we cannot predict when a specific nucleus will decay, we can describe the average behavior of a large number of identical nuclei using concepts like decay constant, half-life, and mean life.

Nuclear Energy

The shape of the binding energy per nucleon curve ($E_{bn}$ vs $A$) reveals that energy can be released from nuclear reactions where nuclei with lower $E_{bn}$ values transform into nuclei with higher $E_{bn}$ values. This corresponds to a net increase in binding energy, and thus a release of energy according to $E = \Delta M c^2$, where $\Delta M$ is the mass difference between the initial and final states.

Nuclear reactions involve energies that are typically millions of times larger than those in chemical reactions (MeV vs eV). This makes nuclear processes a powerful source of energy.

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Fission

Nuclear fission is a process where a heavy, unstable nucleus splits into two or more lighter nuclei, along with the emission of neutrons and a large amount of energy. Fission can occur spontaneously in some heavy isotopes, but it is most commonly induced by bombarding a heavy nucleus with a neutron.

A classic example is the fission of Uranium-235 ($^{235}_{92}\text{U}$) when it absorbs a slow neutron:

$$^1_0\text{n} + ^{235}_{92}\text{U} \rightarrow [^{236}_{92}\text{U}^*] \rightarrow ^{144}_{56}\text{Ba} + ^{89}_{36}\text{Kr} + 3^1_0\text{n} + \text{Energy}$$

The unstable intermediate nucleus ($^{236}_{92}\text{U}^*$) quickly splits into two lighter nuclei (like Barium and Krypton) and typically 2 or 3 neutrons. Other fission products are possible, but they usually fall within the range of mass numbers ($A \approx 90$ to 140) where the binding energy per nucleon is higher than that of Uranium ($A=235$).

The energy released per fission event of a uranium nucleus is typically around 200 MeV. This energy release can be estimated from the binding energy curve: the binding energy per nucleon for a heavy nucleus like Uranium ($A \approx 235$) is about 7.6 MeV, while for the fission fragments ($A \approx 120$), it is about 8.5 MeV. The gain in binding energy per nucleon is $\approx 0.9$ MeV. For 235 nucleons, the total energy released is $\approx 235 \times 0.9 \text{ MeV} \approx 211.5$ MeV. This energy is released as kinetic energy of the fission fragments and neutrons, which is then converted into heat.

The neutrons released during fission can go on to induce fission in other uranium nuclei, leading to a chain reaction. Controlled chain reactions are used in nuclear reactors to generate electricity, while uncontrolled chain reactions result in the explosive energy release of an atomic bomb.

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Nuclear Fusion – Energy Generation In Stars

Nuclear fusion is a process where two or more light nuclei combine or "fuse" to form a single heavier nucleus, releasing a tremendous amount of energy. This energy release occurs because the binding energy per nucleon of the fused, heavier nucleus is greater than the sum of the binding energies per nucleon of the initial lighter nuclei (as indicated by the rise in the binding energy curve for light nuclei).

Examples of fusion reactions involving isotopes of hydrogen:

• $^1_1\text{H} + ^1_1\text{H} \rightarrow ^2_1\text{H} + \text{e}^+ + \nu_e + 0.42 \text{ MeV}$ (Proton-proton fusion)
• $^2_1\text{H} + ^2_1\text{H} \rightarrow ^3_2\text{He} + ^1_0\text{n} + 3.27 \text{ MeV}$ (Deuterium-deuterium fusion)
• $^2_1\text{H} + ^2_1\text{H} \rightarrow ^3_1\text{H} + ^1_1\text{H} + 4.03 \text{ MeV}$ (Deuterium-deuterium fusion)

For fusion to occur, the positively charged nuclei must be brought close enough (within the range of the strong nuclear force, a few fm) to overcome their mutual electrostatic (Coulomb) repulsion. This requires the nuclei to have very high kinetic energies, which means the process requires extremely high temperatures (millions of degrees Celsius). Fusion reactions that occur at such high temperatures are called thermonuclear fusion.

Thermonuclear fusion is the primary energy source in stars, including our Sun. The Sun's core temperature is about $1.5 \times 10^7$ K, which is sufficient for fusion to occur. The main fusion process in the Sun is the proton-proton (p-p) chain, a multi-step cycle that effectively converts four protons into one helium nucleus ($^4_2\text{He}$), positrons, neutrinos, and gamma rays, releasing approximately 26.7 MeV of energy per complete cycle. This process is responsible for the Sun's energy output, powering the light and heat we receive on Earth.

Stars synthesize heavier elements from lighter ones through fusion in their cores. This process continues until elements around the peak of the binding energy curve (like iron) are formed. Fusion of elements heavier than iron would require energy input rather than releasing it.

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Controlled Thermonuclear Fusion

Scientists are researching ways to replicate thermonuclear fusion on Earth in a controlled manner to generate clean and abundant energy. The main challenge is achieving and maintaining the extremely high temperatures ($> 10^8$ K) required for fusion and confining the resulting plasma (a superheated state of matter consisting of ions and electrons) for a sufficient time to allow fusion reactions to occur and produce net energy. Since no physical container can withstand such temperatures, methods like magnetic confinement (using strong magnetic fields to hold the plasma) and inertial confinement (using powerful lasers to rapidly heat and compress a fuel pellet) are being investigated.

Successful development of controlled fusion reactors could potentially provide a long-term, sustainable energy source for humanity, using readily available fuels like deuterium and tritium.

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