A Comprehensive Guide to Modern Physics: Nuclear Physics and Solids

24.1 Nucleus: Discovery of Nucleus

The journey into the atomic nucleus began in 1911 with Ernest Rutherford's groundbreaking gold foil experiment. Before this, J.J. Thomson's 'plum pudding' model proposed a positively charged sphere with electrons embedded within it. Rutherford, along with his students Hans Geiger and Ernest Marsden, fired alpha particles (positively charged helium nuclei) at a thin sheet of gold foil. The key observations were:

• Most alpha particles passed straight through, indicating that atoms are mostly empty space.
• A small fraction of alpha particles were deflected at large angles.
• A very few (about 1 in 8000) were scattered back, implying a very dense, positively charged center.

These results led Rutherford to propose the nuclear model of the atom, where a tiny, dense, positively charged nucleus resides at the center, with electrons orbiting it. This discovery fundamentally changed our understanding of atomic structure.

24.2 Nuclear Density; Mass Number; Atomic Number

The nucleus is an incredibly dense region. Its density is approximately 2.3 x 10¹⁷ kg/m³, far exceeding the density of ordinary matter. This remarkable density arises because almost all of the atom's mass is concentrated in a minuscule volume.

• Mass Number (A): Represents the total number of protons and neutrons (collectively called nucleons) in the nucleus. It is an integer and is approximately equal to the atomic mass in atomic mass units (amu).
• Atomic Number (Z): Represents the number of protons in the nucleus. This number uniquely identifies a chemical element. In a neutral atom, Z is also equal to the number of electrons.
• Number of Neutrons (N): Calculated as A - Z.

A nucleus is often represented as ^A_ZX, where X is the chemical symbol of the element.

Example:

Consider Carbon-12 (¹²₆C). Here, A = 12 (mass number) and Z = 6 (atomic number). This means Carbon-12 has 6 protons and 12 - 6 = 6 neutrons.

24.3 Atomic Mass; Isotopes

• Atomic Mass: The mass of an atom, typically expressed in atomic mass units (amu). One amu is defined as 1/12th the mass of a carbon-12 atom. It's important to distinguish atomic mass (the actual mass of a specific isotope) from the average atomic weight of an element, which is a weighted average of the atomic masses of all its naturally occurring isotopes.
• Isotopes: Atoms of the same element (same atomic number Z) but with different numbers of neutrons (and thus different mass numbers A). Isotopes have identical chemical properties but may have different physical properties (e.g., mass, density, nuclear stability).

Example:

Hydrogen has three common isotopes:

• Protium (¹₁H): 1 proton, 0 neutrons.
• Deuterium (²₁H): 1 proton, 1 neutron (heavy hydrogen).
• Tritium (³₁H): 1 proton, 2 neutrons (radioactive).

24.4 Einstein’s Mass-Energy Relation

One of the most profound equations in physics, E=mc², proposed by Albert Einstein, states the equivalence of mass and energy. This equation implies that mass can be converted into energy, and vice versa. Here:

• E is the energy.
• m is the mass.
• c is the speed of light in a vacuum (approximately 3 x 10⁸ m/s).

Because 'c' is a very large number, even a tiny amount of mass can be converted into an enormous amount of energy. This principle is fundamental to understanding nuclear reactions.

24.5 Mass Defect, Packing Fraction, BE per Nucleon

• Mass Defect (Δm): When protons and neutrons combine to form a nucleus, the mass of the resulting nucleus is always slightly less than the sum of the individual masses of the constituent protons and neutrons. This difference in mass is called the mass defect. It is given by: Δm = (Z * m_p + N * m_n) - m_nucleus, where m_p is the mass of a proton, m_n is the mass of a neutron, and m_nucleus is the actual mass of the nucleus.
• Binding Energy (BE): The mass defect (Δm) is converted into energy according to E=mc². This energy, known as the nuclear binding energy, is the energy required to break a nucleus into its constituent protons and neutrons. It is also the energy released when a nucleus is formed from its constituent nucleons. A larger binding energy indicates a more stable nucleus.
• Binding Energy per Nucleon (BE/A): This is the total binding energy divided by the mass number (A). It's a measure of the stability of a nucleus. Nuclei with higher binding energy per nucleon are more stable. The curve of binding energy per nucleon versus mass number shows that nuclei with mass numbers around 50-60 (like Iron-56) have the highest binding energy per nucleon, indicating maximum stability. This explains why both nuclear fission (heavy nuclei splitting) and nuclear fusion (light nuclei combining) release energy.
• Packing Fraction: Defined as (Mass Defect / Mass Number). It's another way to express the stability of a nucleus, although less commonly used than binding energy per nucleon.

24.6 Creation and Annihilation

• Pair Production (Creation): A high-energy photon (gamma ray) can spontaneously transform into an electron-positron pair when passing near a heavy nucleus. This process demonstrates the conversion of energy into mass, requiring the photon's energy to be at least twice the rest mass energy of an electron (approximately 1.02 MeV). The nucleus is needed to conserve momentum.
• Pair Annihilation: The reverse process, where an electron and its antiparticle, a positron, collide and annihilate each other, converting their entire mass into energy in the form of two gamma-ray photons. This process demonstrates the conversion of mass into energy.

24.7 Nuclear Fission and Fusion, Energy Released

These are two of the most significant nuclear reactions, both releasing immense amounts of energy due to changes in binding energy per nucleon.

• Nuclear Fission: The process where a heavy atomic nucleus splits into two or more smaller nuclei, along with a few neutrons and a large amount of energy. This is typically initiated by bombarding a heavy nucleus (like Uranium-235 or Plutonium-239) with a neutron. The released neutrons can then cause further fission reactions, leading to a chain reaction. This principle is used in nuclear power plants and atomic bombs.
• Nuclear Fusion: The process where two or more light atomic nuclei combine to form a heavier nucleus, releasing a tremendous amount of energy. This process occurs under extremely high temperatures and pressures, such as those found in the core of stars (e.g., the Sun fuses hydrogen into helium). Fusion is considered a potentially cleaner and more abundant energy source than fission, but achieving controlled fusion on Earth remains a significant scientific and engineering challenge.

Energy Released:

In both fission and fusion, the total binding energy of the products is greater than the total binding energy of the reactants. This difference in binding energy is released as kinetic energy of the products and gamma rays, manifesting as a decrease in the total mass (mass defect) which is converted into energy via E=mc².

25.1 Energy Bands in Solids (Qualitative Ideas)

In isolated atoms, electrons occupy discrete energy levels. However, when atoms come together to form a solid, their electron orbitals overlap. According to the Pauli Exclusion Principle, no two electrons can have the same set of quantum numbers. This overlap causes the discrete energy levels of individual atoms to split into a large number of closely spaced energy levels, forming continuous energy bands.

The two most important bands for understanding electrical conductivity are:

• Valence Band: The highest energy band that is completely or partially filled with electrons at absolute zero temperature. These electrons are typically involved in bonding.
• Conduction Band: The next higher energy band, which is usually empty or partially filled. Electrons in this band are free to move throughout the material and contribute to electrical conduction.
• Energy Gap (Band Gap, E_g): The energy difference between the top of the valence band and the bottom of the conduction band. No electron can exist in this region.

25.2 Difference between Metals, Insulators and Semi-conductors using Band Theory

Band theory provides a clear explanation for the vastly different electrical conductivities of materials:

• Metals (Conductors):
  • In metals, the valence band and conduction band either overlap or the conduction band is partially filled.
  • Electrons can easily move into higher energy states within the conduction band with very little energy input (e.g., from thermal energy at room temperature).
  • This abundance of free electrons makes metals excellent electrical conductors.
  • Example: Copper, Aluminum, Gold.
• Insulators:
  • Insulators have a large energy gap (E_g > 3 eV to 5 eV) between a completely filled valence band and an empty conduction band.
  • A significant amount of energy is required for electrons to jump from the valence band to the conduction band.
  • At room temperature, very few electrons have enough energy to make this jump, resulting in very low conductivity.
  • Example: Glass, Rubber, Wood, Diamond.
• Semiconductors:
  • Semiconductors have a smaller energy gap (E_g ≈ 0.5 eV to 2 eV) compared to insulators.
  • At absolute zero, the valence band is full and the conduction band is empty, behaving like an insulator.
  • However, at room temperature, some electrons gain enough thermal energy to jump across the small band gap from the valence band to the conduction band, leaving behind 'holes' in the valence band.
  • Both the electrons in the conduction band and the holes in the valence band contribute to electrical conductivity.
  • Their conductivity increases with increasing temperature.
  • Example: Silicon (Si), Germanium (Ge).

25.3 Intrinsic and Extrinsic Semiconductors

• Intrinsic Semiconductors:
  • Pure semiconductors, without any impurities.
  • At absolute zero, they behave as insulators.
  • At room temperature, electron-hole pairs are created purely by thermal excitation.
  • The number of electrons in the conduction band is equal to the number of holes in the valence band.
  • Their conductivity is low and sensitive to temperature.
  • Example: Pure Silicon, Pure Germanium.
• Extrinsic Semiconductors:
  • Semiconductors whose conductivity has been significantly increased by intentionally adding a small amount of impurities (doping).
  • Doping creates either excess free electrons or excess holes, dramatically increasing conductivity.
  • There are two types of extrinsic semiconductors:
  • n-type semiconductor: Created by doping an intrinsic semiconductor with pentavalent impurities (e.g., Phosphorus, Arsenic) which have 5 valence electrons. Four electrons form covalent bonds, and the fifth electron becomes a 'donor' electron, easily moving to the conduction band. Electrons are the majority charge carriers.
  • p-type semiconductor: Created by doping an intrinsic semiconductor with trivalent impurities (e.g., Boron, Gallium) which have 3 valence electrons. These form 3 covalent bonds, creating a 'hole' in the fourth position, which can accept an electron. Holes are the majority charge carriers.