Semi Empirical Mass Formula - Liquid Drop Model of the Nucleus

SEMF

Semi-Empirical Mass Formula [Bethe-Weizsäcker Formula]

Derivation and Explanation

Statement:

The Semi‑Empirical Mass Formula (SEMF) gives the nuclear binding energy \(B(A,Z)\) of a nucleus with mass number \(A\) and atomic number \(Z\) as the sum of five macroscopic terms derived from the liquid‑drop model:

Formula :

\[B(A,Z)=a_V A - a_S A^{2/3} - a_C \frac{Z(Z-1)}{A^{1/3}} - a_A \frac{(A-2Z)^2}{A} + \delta(A,Z)\]

Sometimes the Coulomb term is written as \(a_C \dfrac{Z^2}{A^{1/3}}\) and the pairing term \(\delta\) has the usual form given below.

Derivation: Physical origin of each term :



Volume term \(a_V A\) :

The nucleus behaves like an incompressible liquid drop. Each nucleon binds with a roughly constant number of nearest neighbours, so binding energy is proportional to \(A\). Hence \(a_V A\) with \(a_V>0\).

Surface term \(-a_S A^{2/3}\) :

Nucleons at the surface have fewer neighbours and so contribute less binding. Surface area scales as \(A^{2/3}\), giving a negative correction \(-a_S A^{2/3}\).

Coulomb term \(-a_C Z(Z-1)/A^{1/3}\) :

Protons repel by Coulomb force. Approximating the nucleus as a uniformly charged sphere of radius \(R\propto A^{1/3}\) leads to electrostatic energy \(\propto Z^2/R \sim Z^2/A^{1/3}\). The factor \(Z(Z-1)\) corrects for self‑interaction.

Asymmetry (or Pauli) term \(-a_A (A-2Z)^2/A\) :

Quantum mechanically, neutrons and protons fill Fermi levels. Minimum energy occurs when \(N\approx Z\) for small nuclei. Deviation from symmetric \(N=Z\) costs kinetic (Fermi) energy, producing a term quadratic in \(N-Z\): since \(N=A-Z\), this becomes \((A-2Z)^2/A\).

Pairing term \(\delta(A,Z)\) :

Because of pairing, nuclei with even numbers of protons and neutrons are extra stable. Empirical form: \[\delta(A,Z)=\begin{cases} +a_P A^{-1/2} & \text{even-}Z,\,\text{even-}N\\ -a_P A^{-1/2} & \text{odd-}Z,\,\text{odd-}N\\ 0 & \text{if } A \text{ is odd}\end{cases}\] This term is small and alternates sign depending on nucleon parity.

Notes on coefficients and units :

Typical fitted values(approximately): \(a_V = 15.5 \, MeV\)

\(a_S = 17.0 \, MeV\)

\(a_C = 0.71 \, MeV\)

\(a_A = 23.0 \, MeV\)

\(a_P = 11.0 \, MeV\)

These are empirical — obtained by fitting measured nuclear masses.

Short derivation sketch for asymmetry term :

Treat protons and neutrons as independent Fermi gases. Fermi energy scales as \(E_F\propto (n)^{2/3}\) where density \(n\) differs when \(N\ne Z\). Expanding the total kinetic energy to second order in the neutron excess gives an energy contribution \(\propto (N-Z)^2/A\), producing the asymmetry term shown above.

Example / application & remarks :

Binding energy per nucleon \(B/A\) predicted by SEMF peaks near \(A\sim 56\), explaining iron‑group stability and why heavy nuclei fission while light nuclei fuse. SEMF also predicts approximate mass parabolae and decay energetics (Q‑values) qualitatively well.

Anomalous Zeeman Effect and Paschen-Back Effect

Anomalous Zeeman Effect

Definition:

The Anomalous Zeeman Effect is the splitting of spectral lines into more than three components when an atom is placed in an external magnetic field, typically observed in atoms having unpaired electron spin and non-zero total angular momentum.

Explanation :

1. In the normal Zeeman effect only orbital angular momentum is considered and energy levels split into three (a triplet) because of simple magnetic interaction.
2. Most atoms, however, have both orbital L and spin S angular momenta which couple to give total angular momentum J. The interaction with the magnetic field depends on the Landé g-factor (g_J), not just on orbital motion.
3. Each atomic level splits into (2J + 1) magnetic sublevels labelled by m_J, producing multiple components in the observed spectral lines. Selection rules (Δm_J = 0, ±1) determine which transitions are allowed, giving a complex pattern of π and σ lines.
4. The anomalous effect thus reveals the role of electron spin and the vector coupling of L and S, and was historically important evidence for the existence of electron spin and for quantum theory.

Key formula:

\[\Delta E = \mu_B\, g_J\, B\, m_J\]

where \(\mu_B\) is the Bohr magneton, \(B\) is the magnetic field, \(g_J\) is the Landé g-factor, and \(m_J\) is the magnetic quantum number.

Paschen Back Effect :

In very strong magnetic fields, the coupling between orbital angular momentum \(L\) and spin \(S\) breaks down. The magnetic interaction becomes dominant compared to spin orbit coupling, causing \(L\) and \(S\) to align independently with the field. This leads to a simpler splitting pattern than the anomalous Zeeman effect, similar to the normal Zeeman effect.

This occurs When magnetic energy \(\mu_B B\) is much greater than spin orbit interaction energy.

Electric Field Visualizer

Electric Field Visualizer

By KAZ

Charge magnitude

1.0

Grid density

20

Field scale

1.0

Color map Hot (red → yellow) Viridis-like Blue-Red

Charges (drag to move; click to remove)

Arrows are unit vectors (direction only). Their color encodes the magnitude of the electric field at that point. Charges glow and scale with magnitude.Charges can be dragged around using mouse.

Threshold Energy For Nuclear Reaction

Threshold Energy in Nuclear Reactions

In nuclear physics, the threshold energy is the minimum kinetic energy that a projectile particle must possess in order to make a particular nuclear reaction occur.

Definition

The threshold energy for a nuclear reaction is defined as the minimum energy of the incoming particle required to overcome the energy difference between the reactants and the products, ensuring the reaction can proceed while conserving both energy and momentum.

Explanation

If a reaction absorbs energy (i.e., it is endothermic), the projectile must supply not only this reaction energy but also additional energy to satisfy momentum conservation. Thus, the actual threshold energy is slightly higher than the reaction’s Q-value (when Q is negative).

Threshold Energy (E_th) = −Q × (1 + m_a / M_A)

where Q is the reaction energy, m_a is the mass of the projectile, and M_A is the mass of the target nucleus.

Example

In the reaction p + ³H → ³He + n, if the reaction Q-value is negative, the proton must have at least the threshold energy computed from the above relation for the reaction to take place.

Summary

The concept of threshold energy helps determine the minimum energy requirement for initiating endothermic nuclear reactions and plays a vital role in nuclear reactor design and particle accelerator physics.

Isospin - Explanation and Defintion

Isospin (Isotopic Spin) in Nuclear Physics

Isospin or isotopic spin is a fundamental concept in nuclear and particle physics that helps explain the strong interaction between nucleons — protons and neutrons. The term was introduced by Werner Heisenberg in 1932 to describe the striking similarity in the behavior of protons and neutrons under the strong nuclear force.

Definition

Isospin may be defined as a quantum number that treats the proton and neutron as two different states of the same particle (the nucleon), differing only by their electric charge. It is analogous in mathematics to ordinary spin, but it does not represent physical rotation — rather, it represents an internal symmetry related to the strong force.

Concept and Analogy

The idea behind isospin is that the strong nuclear force does not depend on electric charge — it acts equally on protons and neutrons. Thus, they can be considered as two members of an isospin doublet:

Particle	Isospin (I)	Isospin Projection (I3)
Proton (p)	1/2	+1/2
Neutron (n)	1/2	−1/2

Here, the total isospin \( I = \frac{1}{2} \) for both particles, while the component \( I_3 \) distinguishes between them. This analogy closely resembles spin-up and spin-down states in quantum mechanics.

Isospin Multiplets

Isospin symmetry extends beyond nucleons to other hadrons that experience the strong force. Particles that differ only in charge but have similar masses and interactions form isospin multiplets. For example:

• The three pions (π⁺, π⁰, π⁻) form an isospin triplet with \( I = 1 \).
• The nucleons (proton and neutron) form an isospin doublet with \( I = 1/2 \).

Significance of Isospin

The concept of isospin is extremely useful in simplifying nuclear and particle physics problems:

• It explains why the strong nuclear force is nearly charge-independent.
• It allows classification of particles into groups (multiplets) with similar strong interaction properties.
• It provides a symmetry principle that was later generalized in the framework of SU(2) and SU(3) symmetry groups.

Conclusion

In summary, isospin is not a physical spin but an abstract quantum number representing a symmetry between protons and neutrons under the strong nuclear force. It remains a cornerstone in understanding hadronic interactions and the classification of subatomic particles.

Primary Vs Secondary Cosmic Rays

Primary and Secondary Cosmic Rays

Cosmic rays are high-energy particles originating from outer space that constantly strike the Earth's atmosphere. They are broadly classified into two types — Primary Cosmic Rays and Secondary Cosmic Rays — based on their origin and interaction with the atmosphere.

1. Primary Cosmic Rays

Primary cosmic rays are the original high-energy particles that come directly from outer space before interacting with Earth's atmosphere. They mainly consist of protons (about 90%), alpha particles (about 9%), and a small fraction of heavier nuclei and electrons.

These rays originate from energetic astrophysical sources such as the Sun, supernovae, neutron stars, and distant galaxies. When they enter the Earth's atmosphere, they collide with atomic nuclei, producing a cascade of new particles — giving rise to secondary cosmic rays.

2. Secondary Cosmic Rays

Secondary cosmic rays are produced when primary cosmic rays interact with atoms in the Earth's atmosphere. These collisions generate a shower of new particles such as pions, muons, electrons, neutrinos, and gamma rays.

Many of these secondary particles decay rapidly, but some (like muons) can reach the Earth’s surface and even penetrate underground detectors. Thus, most of the cosmic radiation detected at sea level comes from secondary cosmic rays.

Comparison Table

Property	Primary Cosmic Rays	Secondary Cosmic Rays
Origin	From outer space (solar or galactic sources)	Produced in Earth’s atmosphere
Composition	Mainly protons and alpha particles	Muons, electrons, pions, gamma rays, etc.
Energy	Extremely high (up to 1020 eV)	Lower than primary cosmic rays
Atmospheric Interaction	No interaction before entering atmosphere	Formed after collisions in the atmosphere
Detection	Detected outside the atmosphere (via satellites)	Detected on or near Earth's surface

In essence, primary cosmic rays are the original messengers from space, while secondary cosmic rays are their atmospheric offspring. Understanding both types helps physicists study particle interactions and trace the energetic processes occurring across the universe.

Cosmic Rays - Latitude and Longitude Effect

Latitude and Altitude Effects on Cosmic Rays

Latitude Effect

Definition: Latitude effect may be defined as the effect that shows the intensity of the cosmic rays with the geometrical latitude. It shows that the intensity of the cosmic rays is maximum at the poles where geometrical latitude is and is minimum at the equator where the geometrical latitude is.

The Earth's magnetic field is the main reason for the decrease in cosmic ray intensity at the equator. In the poles, the charged particles are travelling parallel to the direction of the magnetic field. Thus, they can travel to Earth almost unhindered, so the intensity is always maximum at the poles. But when we consider the scenario of the equator, the charged particles have to travel in a perpendicular direction to the field and face the maximum hindrance. Only the particles having enough energy to cut through this barrier can reach the equator, thus we get minimum intensity at the equator.

In summary, cosmic ray intensity increases with latitude and is maximum at the poles due to the Earth's magnetic field orientation.

Altitude Effect

Definition: It may be defined as the effect which shows the variation of the cosmic rays intensity with the altitude (height). The cosmic rays intensity increases with the increase of the altitude and is maximum when we reach an altitude of about 20 km. If we further increase the altitude then the intensity of the cosmic rays decreases. This effect is called the altitude effect.

The latitude effect is generally known as the change of the physical quantity with change in latitude whereas the altitude effect is the change of the physical quantity with respect to the change in the height.

Thus, while the latitude effect depends on geomagnetic influence, the altitude effect reflects atmospheric absorption and particle generation processes.

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Properties of Thermal Neutrons Vs Fast Neutrons

Property	Thermal Neutrons	Fast Neutrons
Definition	Neutrons in thermal equilibrium with surroundings (low kinetic energy).	High kinetic energy neutrons emitted from nuclear reactions.
Typical Energy Range	≈ 0.025 eV (at ~300 K)	≈ 1 MeV – 10 MeV (or higher)
Speed	≈ 2,200 m/s	≈ 10⁷ m/s
Source	Produced when fast neutrons are slowed by a moderator (water, graphite, heavy water).	Produced directly from fission or other nuclear reactions.
Interaction with Nuclei	High probability of absorption and inducing fission in certain isotopes (e.g., U-235).	Lower absorption probability; more likely to cause scattering (elastic/inelastic).
Use in Reactors	Used in thermal reactors (with moderators) to sustain chain reactions.	Used in fast reactors/breeder reactors (no moderator) for breeding and fast-spectrum reactions.
Cross-section (Interaction Probability)	Generally higher absorption and fission cross-sections for many fissile nuclei.	Generally lower absorption cross-sections; scattering dominates.

Interactive Dipole Wave Interference - A Shader Animation

An animation of dipole wave interference field. Press Space to pause or play the animation. If one looks carefully at where the wavefield vanishes one can see that they waves have a path difference of half a wavelength or an odd integral multiples of it there. One can also note that far-field pattern is different from the near-field pattern. It can be observed that at a distance of half-wavelength the sources when they are in phase as shown interfere destructively along the dipole axis and constructively normal to it.

Q Value Equation for a General Binary Nuclear Reaction

 

Simulation of Proton Scattering by the Coulomb Potential of a Heavy Nucleus - Rutherford Scatering

Scattering Controls

Projectile Energy 150

Impact parameter (px) 40

Nuclear charge Z 50

Coulomb constant k 10

Speed 3

Last scattering angle: —

Histogram (angle counts)

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Spectral Terms and Notations

Spectral Terms and Notation in Atomic Physics

Spectral terms provide a compact way of describing the quantum states of electrons in atoms. They summarize the total orbital angular momentum, spin, and multiplicity of a given electronic configuration.

General Form of a Spectral Term

A spectral term is written as:

$$^{2S+1}L_J$$

• S: Total electron spin quantum number.
• 2S+1: Multiplicity (singlet, doublet, triplet, etc.).
• L: Total orbital angular momentum, denoted by letters (S, P, D, F, ... for L=0,1,2,3,...).
• J: Total angular momentum, combining L and S, ranging from \( \lvert L+S \rvert \) to \( \lvert L-S \rvert \) in steps of 1.

Examples of Spectral Terms

(a) Hydrogen Ground State

The electron has spin \(S = \tfrac{1}{2}\) and orbital angular momentum \(L = 0\). Thus:

$$^{2}S_{1/2}$$

This represents a doublet-S state with total \(J = 1/2\).

(b) Helium \(1s2s\) Configuration

For two electrons, spins can pair to form either:

• Singlet state: \(S=0\), term \(^1S_0\)
• Triplet state: \(S=1\), term \(^3S_1\)

This splitting explains why helium exhibits both singlet and triplet spectral series.

Notes on Multiplicity

The multiplicity \(2S+1\) determines how many closely spaced energy levels appear. Higher multiplicity (like triplets) often correspond to lower energy due to electron exchange effects.

In short, spectral term notation provides a compact way to describe the structure of atomic energy levels.

Motion of a Block On an Inclined Plane

mg

mg sinθ

mg cosθ / N

f

Inclined Plane — Controls

Angle θ 30°

tanθ = 0.577

Material pair (μ_s, μ_k)

Wood on Wood — μs=0.4 μk=0.2 Ice on Ice — μs=0.1 μk=0.03 Rubber on Concrete — μs=1.0 μk=0.8 Steel on Steel — μs=0.8 μk=0.6 Teflon on Steel — μs=0.04 μk=0.02 Glass on Glass — μs=0.94 μk=0.4

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Static Friction: \( f_s \leq \mu_s N = \mu_s m g \cos\theta \)

Kinetic Friction: \( f_k = \mu_k N = \mu_k m g \cos\theta \)

Net Force (parallel): \( F_{\text{net}} = m g \sin\theta - f \)

Condition for motion: \( \tan\theta > \mu_s \)

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