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MPH-017 Solved Assignment January 2026 | NUCLEAR AND PARTICLE PHYSICS | IGNOU MSCPH

Q1.a. Calculate the mass defect of Boron-10 and proton separation energy. Given: M (10B) = 10.0129369 u M(Be) = 9.0121831u M (Proton) = 1.0078250 u

Key Points:

Mass defect is the difference between individual nucleon masses and actual nuclear mass.
A positive mass defect indicates nuclear stability and binding energy.
Proton separation energy is the energy required to remove one proton.
Sp is calculated from the mass difference of parent nucleus, daughter nucleus, and proton.

Answer: In nuclear physics, the mass defect and separation energy are fundamental concepts related to the stability and binding forces within an atomic nucleus. The mass defect quantifies the difference between the total mass of individual nucleons and the actual measured mass of the nucleus, while the proton separation energy represents the minimum energy required to remove a single proton from a nucleus. To calculate the mass defect of Boron-10 (¹⁰B), we first need to determine the total mass of its ...

Q1.b. The mass defects for 7 Li and 240 Pu are 0.040 u, and 1.9324 u respectively. Calculate the corresponding atomic mass.

Key Points:

Mass defect (Δm) is the difference between constituent nucleon masses and actual atomic mass.
Formula for atomic mass: M_atom = (Z × m_H + N × m_n) - Δm.
Z is atomic number, N is neutron number (A-Z).
Use standard values for hydrogen atom mass (m_H ≈ 1.007825 u) and neutron mass (m_n ≈ 1.008665 u).

Answer: In nuclear physics, the mass defect (Δm) is a fundamental concept that quantifies the difference between the sum of the individual masses of the constituent nucleons (protons and neutrons) in a nucleus and the actual measured mass of the atom. This difference in mass is directly related to the nuclear binding energy via Einstein's mass-energy equivalence principle (E=mc²), as detailed in Block 2, Unit 5 of the MPH-017 course material. To calculate the atomic mass (M_atom) when the mass defect (...

Q1.c. For the nuclear charge density Pch = Po e-gor, compute the charge form factor and the mean square charge radius (r2).

Key Points:

Nuclear charge density `Pch = Po e⁻ᵍᵒʳ` must be normalized to total charge `Ze`.
Normalization yields `Po = Ze * (g₀)³ / (8π)` for `Pch(r)`.
Charge form factor `F(q)` is Fourier Transform of normalized charge density.
The derived charge form factor is `F(q) = (g₀)⁴ / (g₀² + q²)²`.

Answer: The nuclear charge density `Pch(r)` is a fundamental quantity describing the distribution of charge within an atomic nucleus. The charge form factor `F(q)` and the mean square charge radius `<r²>` are derived from this density and provide insights into the nuclear structure, typically determined through electron scattering experiments. For the given nuclear charge density `Pch = Po e⁻ᵍᵒʳ`, we proceed to compute the charge form factor and the mean square charge radius. **1. Normalization of the...

Q1.d. Explain and justify the observed deviations in magnetic dipole moment and quadrupole moment of deuteron.

Key Points:

Deuteron's pure S-state model predicts μ_d = 0.880 μ_N and Q_d = 0.
Observed magnetic moment μ_d = 0.8574 μ_N, showing a deviation.
Observed electric quadrupole moment Q_d = +0.282 fm², indicating non-spherical shape.
Deviations signify deuteron is not a pure S-state, but a mix of S and D states.

Answer: The deuteron, composed of a proton and a neutron, is the simplest bound nuclear system. Its observed magnetic dipole moment and electric quadrupole moment deviate significantly from predictions based on a simple model, providing crucial insights into the nature of the strong nuclear force and the deuteron's internal structure. Initially, if the deuteron were assumed to be in a pure S-state (L=0), its total angular momentum (I=1) would arise solely from the parallel alignment of the proton's and...

Q1.e. List any three major differences between low energy n-p scattering and p-p scattering. Calculate the third component of isospin T3 for 35/17 Cl and 32/16 S.

i) List any three major differences between low energy n-p scattering and p-p scattering. (300 words)
ii) Calculate the third component of isospin T3 for 35/17 Cl and 32/16 S. (200 words)

Key Points:

p-p scattering involves repulsive Coulomb force, absent in n-p scattering.
n-p system can be in T=0 or T=1 isospin states.
p-p system is restricted to the T=1 isospin state due to Pauli exclusion principle.
Nuclear scattering lengths for n-p (triplet/singlet) differ significantly from p-p.

Answer: Low-energy nuclear scattering experiments provide crucial insights into the strong nuclear force, revealing fundamental differences between various nucleon-nucleon interactions. The scattering of neutrons off protons (n-p scattering) and protons off protons (p-p scattering) are two such fundamental processes. While both are governed by the strong nuclear force, the presence or absence of the electromagnetic (Coulomb) interaction, along with quantum mechanical principles like the Pauli exclusion ...

Q2.a. Write the semi-empirical mass formula. On the basis of semi-empirical mass formula, derive the condition of stability of a given nucleus.

Key Points:

Semi-empirical mass formula (SEMF) approximates nuclear binding energy based on the liquid drop model.
SEMF includes volume, surface, Coulomb, asymmetry, and pairing energy terms.
Nuclear stability for a given A corresponds to the minimum nuclear mass M(A,Z).
Stable Z₀ is found by differentiating M(A,Z) with respect to Z and setting it to zero.

Answer: The semi-empirical mass formula (SEMF), also known as the Weizsäcker formula, is a theoretical expression used to approximate the binding energy and mass of an atomic nucleus based on the liquid drop model. It provides a good estimate for the binding energy of most nuclei, describing how nuclear forces contribute to the stability of a nucleus. **Semi-Empirical Mass Formula** The binding energy B(A,Z) of a nucleus with mass number A and atomic number Z is given by the formula: B(A,Z) = avA - ...

Q2.b. Determine the expected shell-model quadrupole moment of 209Bi (9/2).

Key Points:

209Bi is an odd-proton nucleus (Z=83) with a closed neutron shell (N=126).
The given spin J=9/2⁺ implies the valence proton is in a 1g9/2 orbital (l=4, j=9/2).
Shell-model Q for j = l + 1/2 proton: Q = - (2j - 1) / (2j + 2) × <r²>.
Nuclear radius R ≈ 1.2 fm × A^(1/3) and <r²> ≈ (3/5)R² for estimation.

Answer: To determine the expected shell-model quadrupole moment of ²⁰⁹Bi (9/2), we first need to identify the properties of the nucleus and its valence nucleon, then apply the appropriate shell-model formula. Bismuth-209 (²⁰⁹Bi) has 83 protons (Z=83) and 126 neutrons (N=209-83). Since N=126 is a magic number, the neutron core is closed, and its contribution to the nuclear spin and quadrupole moment is negligible. Therefore, the nuclear properties are primarily determined by the single odd proton (the 8...

Q2.c. Write short notes on

i) Internal conversion, (167 words)
ii) Nuclear isomerism (167 words)
iii) ẞ-decay process with one example. (167 words)

Key Points:

Internal conversion: Nucleus ejects atomic electron directly, competing with gamma emission, no photon involved.
ICE energy calculation: E_e = E_γ - B_e (nuclear transition energy minus electron binding energy).
Nuclear isomerism: Long-lived excited nuclear states due to large spin differences, causing forbidden transitions.
Isomeric transition (IT): Decay from an isomer; ⁹⁹mTc is a medical imaging example.

Answer:

Q2.d. Obtain an expression of momentum distribution of ẞ-particles in the momentum range Pẞ to (PB + dpẞ), considering the Coulomb correction term. Also, derive the energy distribution function to get the shape of the ẞ-decay spectrum.

Key Points:

Beta-particle momentum distribution originates from Fermi's Golden Rule and available phase space.
The initial momentum distribution is proportional to Pβ² dPβ for electron and Pν² dPν for neutrino.
Coulomb correction term F(Z, Pβ) accounts for electrostatic interaction with the daughter nucleus.
Momentum distribution: dN/dPβ ∝ F(Z, Pβ) * Pβ² * (E₀ - Eβ)².

Answer: The distribution of beta-particles in nuclear beta decay is a fundamental aspect governed by Fermi's theory, which utilizes the Golden Rule of time-dependent perturbation theory. This theory states that the transition probability per unit time (λ) is proportional to the square of the interaction matrix element and the density of final states. For allowed beta decays, the nuclear matrix element (|Mfi|²) is considered relatively constant and independent of lepton momenta. To obtain the momentum d...

Q2.e. Calculate the energy produced (in Joules) by fission of 10 g of 235U. Given: The energy released per fission of 235U is 200 MeV.

Key Points:

Nuclear fission is the process where heavy nuclei split, releasing significant energy.
Uranium-235 (²³⁵U) is a key fissile isotope, releasing 200 MeV per fission event.
First, calculate moles of ²³⁵U using its mass (10 g) and molar mass (235 g/mol).
Convert moles to the number of atoms/fissions using Avogadro's number (6.022 × 10²³ atoms/mol).

Answer: The energy produced by the nuclear fission of a given mass of a fissile material like Uranium-235 (²³⁵U) is a critical calculation in nuclear and particle physics, forming a core part of the MPH-017 course discussions on nuclear energy. Nuclear fission is a process where a heavy atomic nucleus splits into smaller fragments, typically after absorbing a neutron, releasing a substantial amount of energy due to the mass defect converted into energy as per Einstein's mass-energy equivalence principle...

Q3.a. What is reciprocity theorem of nuclear reactions? Obtain its mathematical expression. Rewrite the reciprocity theorem considering the spin of all the involved nuclei.

Key Points:

Reciprocity theorem relates cross-sections of a nuclear reaction and its inverse.
It stems from time-reversal invariance (micro-reversibility) of nuclear forces.
Mathematical expression (no spin): p_AB² * dσ_f/dΩ = p_CD² * dσ_r/dΩ.
p_AB and p_CD are relative momenta in the center-of-mass frame.

Answer: The reciprocity theorem, also known as the principle of detailed balance or micro-reversibility, is a fundamental concept in nuclear reaction theory. It establishes a relationship between the cross-section of a nuclear reaction and the cross-section of its inverse reaction. This theorem is a direct consequence of the time-reversal invariance of the strong nuclear force, meaning that the microscopic processes of nuclear interactions are symmetric under reversal of time. Specifically, if a react...

Q3.b. A reactor operating at a power level of 1000 MW uses thermal neutrons producing fissions of 235U. If the thermal flux is 1016m-2s-1, calculate the mass of 235U present in the reactor? Assume of = 580 b.

Key Points:

Reactor power is the total energy released per second from fissions.
Energy per 235U fission is approximately 200 MeV (3.204 × 10⁻¹¹ J).
Total fission rate (fissions/s) equals reactor power divided by energy per fission.
Fission rate also equals (Total 235U atoms) × (fission cross-section) × (neutron flux).

Answer: To calculate the mass of 235U present in a nuclear reactor operating at a specific power level, we need to establish the relationship between reactor power, the rate of fission, the energy released per fission, the thermal neutron flux, and the microscopic fission cross-section. This problem applies fundamental principles of reactor physics, as discussed in the MPH-017 course material concerning nuclear chain reactions and power generation. First, we determine the energy released per 235U fissi...

Q3.c. Explain the phenomenon of nuclear fission. What is the importance of chain reactor in nuclear fission? Explain briefly the principle of operation of a nuclear reactor.

Key Points:

Nuclear fission is the splitting of a heavy nucleus into lighter nuclei, releasing energy and neutrons.
Fission is initiated by neutron absorption, typically in Uranium-235 or Plutonium-239.
The 'mass defect' in fission is converted into substantial energy (E=mc²).
A chain reaction occurs when released neutrons cause further fission events, making it self-sustaining.

Answer: ## Nuclear Fission Nuclear fission is a nuclear reaction in which a heavy nucleus, typically uranium-235 or plutonium-239, splits into two or more smaller nuclei, often called fission products. This process is usually initiated by the absorption of a neutron. When a heavy nucleus absorbs a slow (thermal) neutron, it becomes unstable and cleaves into lighter fragments. This splitting process releases a substantial amount of energy, primarily in the form of kinetic energy of the fission products...

Q3.d. A thin Li target of 0.01 mm thickness is bombarded with a proton beam resulting in 10⁹ neutron/s. Calculate the reaction cross-section σ for this nuclear reaction. Given: Density of Li = 500 kg m⁻³.

Key Points:

Reaction rate (R) depends on incident beam current (I), target number density (n), target thickness (x), and reaction cross-section (σ).
Formula: R = I * n * σ * x is fundamental for calculating cross-sections from observed reaction rates.
Number density (n) is calculated using target material's density, molar mass, and Avogadro's number.
Incident proton beam current (I) is essential for a unique calculation of the cross-section.

Answer: To calculate the reaction cross-section (σ) for the nuclear reaction, we utilize the fundamental relationship between the observed reaction rate, incident particle flux, and target properties. The reaction rate (R) is directly proportional to the incident beam current (I), the number density of target nuclei (n), the target thickness (x), and the reaction cross-section (σ). The formula governing this relationship is R = I * n * σ * x. Here, R represents the number of reactions per second, which...

Q3.e. Define Q-value of a nuclear reaction. Calculate the Q-value of the reaction: 14N(α, ρ) 17O. Given: M (14N) = 14.00753 u M(He) = 4.00386 u M(1H) = 1.00813 u M (17O) = 17.0045 u Whether the reaction is exoergic or endoergic?

Key Points:

Q-value is the energy released (positive) or absorbed (negative) in a nuclear reaction.
Q = (ΣM_reactants - ΣM_products) × c², where M is the rest mass.
A positive Q-value signifies an exoergic reaction; energy is released.
A negative Q-value indicates an endoergic reaction; energy is absorbed.

Answer: In nuclear physics, the Q-value of a nuclear reaction is defined as the amount of energy released or absorbed during the reaction. It represents the difference between the total kinetic energy of the products and the total kinetic energy of the reactants, or equivalently, the difference in the total rest mass energy between the reactants and the products. According to the principles outlined in MPH-017, the Q-value is fundamentally derived from the mass-energy equivalence principle (E=mc²). For...

Q4.a. Check if isospin is conserved in the following reactions:

i) π⁺ + n → π⁰ + p⁺ (300 words)
ii) K⁻ + p⁺ → π⁰ + Λ⁰ (300 words)

Key Points:

Isospin (I) and its third component (I₃) are quantum numbers conserved in strong interactions.
For a reaction to conserve isospin, both total I₃ and possible total I values must be conserved.
The third component of isospin (I₃) is an additive quantum number for reacting particles.
Total isospin (I) for two particles ranges from |I₁ - I₂| to |I₁ + I₂| in integer steps.

Answer: Isospin is a quantum number associated with the strong interaction, analogous to spin. For isospin to be conserved in a reaction, both the total isospin (I) and its third component (I₃) must be conserved. This conservation holds true for strong interactions. The procedure involves calculating the initial and final sums of I₃ values and the range of possible total I values for both sides of the reaction.

Q4.b. Explain why we must define an intrinsic parity for each particle.

Key Points:

Parity (P) is a spatial inversion symmetry, yielding eigenvalues ±1.
Orbital parity is given by (-1)^L, insufficient for fundamental particles.
Intrinsic parity is a fundamental quantum number unique to each particle type.
Total parity is the product of intrinsic and orbital parities.

Answer: Parity (P) is a fundamental discrete symmetry operation in quantum mechanics, representing spatial inversion where coordinates transform as (x, y, z) → (-x, -y, -z). When applied to a particle's wavefunction ψ(r), the parity operator P yields Pψ(r) = ψ(-r). For a system to possess definite parity, its wavefunction must be an eigenfunction of this operator, leading to eigenvalues of either +1 (even parity) or -1 (odd parity). For a particle in an orbital state, its orbital parity is determined b...

Q4.c. From the isospin and hypercharge assignment show that the t and b quarks have charges + 2/3 and - 1/3 respectively.

Key Points:

Gell-Mann-Nishijima formula: Q = I₃ + Y/2 defines particle charge.
Quarks are fundamental constituents, their charges derived from quantum numbers.
Top (t) quark is an 'up-type' quark; assigned I₃ = +1/2 and Y = +1/3.
Bottom (b) quark is a 'down-type' quark; assigned I₃ = -1/2 and Y = +1/3.

Answer: In the realm of particle physics, quarks are fundamental fermions that constitute hadrons. Their properties, such as electric charge (Q), are determined by quantum numbers like isospin (I₃) and hypercharge (Y). The relationship between these quantities is precisely defined by the Gell-Mann-Nishijima formula, a cornerstone of the quark model. According to the Gell-Mann-Nishijima formula, the electric charge (Q) of a particle is given by: Q = I₃ + Y/2 Here, I₃ represents the third component of...

Q4.d. Write down the wave function for π⁺. Use the isospin raising operator to derive the wavefunctions for π (π⁰) and (π⁺).

Key Points:

Pions are isospin I=1 mesons, with π⁺, π⁰, π⁻ corresponding to I₃ = +1, 0, -1.
The π⁺ wave function is |π⁺⟩ = |u anti-d⟩, representing its quark-antiquark composition.
Isospin lowering operator I⁻ acts as I⁻|I, I₃⟩ = √[I(I+1) - I₃(I₃-1)] |I, I₃-1⟩.
Isospin action on quarks: I⁻|u⟩ = |d⟩; on antiquarks: I⁻|anti-d⟩ = |anti-u⟩.

Answer: Pions (π mesons) are pseudo-scalar mesons with a total isospin of I=1, forming an isospin triplet. Each pion state (π⁺, π⁰, π⁻) corresponds to a specific isospin projection I₃. Their wave functions describe their constituent quark-antiquark composition in terms of isospin eigenstates. **Wave Function for π⁺** The π⁺ meson has an isospin projection of I₃ = +1. Its quark composition consists of an up quark (u) and an anti-down quark (anti-d). To consistently describe the isospin of mesons, we as...

Q4.e. Consider K meson of mass mk decaying into two pions of mass mπ each, in the laboratory frame. Show that the energy of each pion is equal to mk / 2 and the momenta are pπ = 1/2 sqrt(mk^2 - 4mπ^2).

Key Points:

Conservation of energy and momentum are fundamental principles in particle decay.
For K meson at rest, initial momentum is zero (P_K = 0) and energy is E_K = mk.
Two-body decay from rest implies pions move in opposite directions with equal momentum magnitudes (pπ).
Energy conservation leads to E_K = 2Eπ, thus Eπ = mk / 2 for each pion.

Answer: To demonstrate the energy and momentum of the pions resulting from the decay of a K meson at rest in the laboratory frame, we must apply the fundamental principles of relativistic kinematics, specifically the conservation of energy and momentum, along with the relativistic energy-momentum relation, as typically covered in MPH-017. This scenario is a classic example of a two-body decay from a parent particle initially at rest. First, we establish the initial state. The K meson, with mass mk, is ...

MPH-017 Solved Assignment — January 2026 / July 2026

NUCLEAR AND PARTICLE PHYSICS | IGNOU Master of Science (Physics) Programme (MSCPH)

MPH-017—January 2026 / July 2026

NUCLEAR AND PARTICLE PHYSICS

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MPH-017 Assignment Questions — January 2026

Q1.a. Calculate the mass defect of Boron-10 and proton separation energy. Given: M (10B) = 10.0129369 u M(Be) = 9.0121831u M (Proton) = 1.0078250 u

Q1.b. The mass defects for 7 Li and 240 Pu are 0.040 u, and 1.9324 u respectively. Calculate the corresponding atomic mass.

Q1.c. For the nuclear charge density Pch = Po e-gor, compute the charge form factor and the mean square charge radius (r2).

Q1.d. Explain and justify the observed deviations in magnetic dipole moment and quadrupole moment of deuteron.

Q1.e. List any three major differences between low energy n-p scattering and p-p scattering. Calculate the third component of isospin T3 for 35/17 Cl and 32/16 S.

Q2.a. Write the semi-empirical mass formula. On the basis of semi-empirical mass formula, derive the condition of stability of a given nucleus.

Q2.b. Determine the expected shell-model quadrupole moment of 209Bi (9/2).

Q2.c. Write short notes on

Q2.d. Obtain an expression of momentum distribution of ẞ-particles in the momentum range Pẞ to (PB + dpẞ), considering the Coulomb correction term. Also, derive the energy distribution function to get the shape of the ẞ-decay spectrum.

Q2.e. Calculate the energy produced (in Joules) by fission of 10 g of 235U. Given: The energy released per fission of 235U is 200 MeV.

Q3.a. What is reciprocity theorem of nuclear reactions? Obtain its mathematical expression. Rewrite the reciprocity theorem considering the spin of all the involved nuclei.

Q3.b. A reactor operating at a power level of 1000 MW uses thermal neutrons producing fissions of 235U. If the thermal flux is 1016m-2s-1, calculate the mass of 235U present in the reactor? Assume of = 580 b.

Q3.c. Explain the phenomenon of nuclear fission. What is the importance of chain reactor in nuclear fission? Explain briefly the principle of operation of a nuclear reactor.

Q3.d. A thin Li target of 0.01 mm thickness is bombarded with a proton beam resulting in 10⁹ neutron/s. Calculate the reaction cross-section σ for this nuclear reaction. Given: Density of Li = 500 kg m⁻³.

Q3.e. Define Q-value of a nuclear reaction. Calculate the Q-value of the reaction: 14N(α, ρ) 17O. Given: M (14N) = 14.00753 u M(He) = 4.00386 u M(1H) = 1.00813 u M (17O) = 17.0045 u Whether the reaction is exoergic or endoergic?

Q4.a. Check if isospin is conserved in the following reactions:

Q4.b. Explain why we must define an intrinsic parity for each particle.

Q4.c. From the isospin and hypercharge assignment show that the t and b quarks have charges + 2/3 and - 1/3 respectively.

Q4.d. Write down the wave function for π⁺. Use the isospin raising operator to derive the wavefunctions for π (π⁰) and (π⁺).

Q4.e. Consider K meson of mass mk decaying into two pions of mass mπ each, in the laboratory frame. Show that the energy of each pion is equal to mk / 2 and the momenta are pπ = 1/2 sqrt(mk^2 - 4mπ^2).