51Nuclear physics

📚
NUCLEAR PHYSICS
▢ Nucleus:

Table 1: Basic Nuclear Facts

Fact
Answer
Charge of nucleus
Positive
Constituents
Protons + neutrons
Collective name
Nucleons
Nuclear radius order
\(10^{-15}m\)
Atomic radius order
\(10^{-10}m\)
Atomic number
\(Z=\) number of protons
Mass number
\(A=\) number of nucleons
Number of neutrons
\(N=A-Z\)
Neutral atom
Protons = electrons = \(Z\)
▢ Nuclear Size:

Table 1: Nuclear Radius

Quantity
Formula / Value
Radius of nucleus
\(R=R_0A^{1/3}\)
Nuclear radius constant
\(R_0\approx1.3\ fermi\)
1 fermi
\(10^{-15}m\)
Radius relation
\(R\propto A^{1/3}\)
Graph
\(\log R\) vs \(\log A\) is straight line
❖ Important Point: Nuclei of different elements have different sizes
▢ Nuclear Density:

Table 1: Nuclear Density Formulae

Quantity
Formula / Value
Density
\(\rho=\frac{Mass\ of\ nucleus}{Volume\ of\ nucleus}\)
Using radius
\(\rho=\frac{mA}{\frac{4}{3}\pi R_0^3A}\)
Result
\(\rho=\frac{3m}{4\pi R_0^3}\)
Dependence on mass number
Independent of \(A\)
Value
\(2.98\times10^{17}kgm^{-3}\)
Order
\(10^{17}kgm^{-3}\)
❖ Important Point: Nuclear density is nearly same for all nuclei
▢ Atomic Mass Unit:
❖ Definition: One atomic mass unit is \(\frac{1}{12}\) of mass of one \(_6C^{12}\) atom

Table 1: Atomic Mass Unit

Quantity
Value
1 amu
\(1.67\times10^{-27}kg\)
1 amu
\(931MeV\)
Energy from mass defect in amu
\(E=931\Delta m\ MeV\)
Energy from mass defect in kg
\(E=\Delta mc^2\)
▢ Types of Nuclei:

Table 1: Isotopes, Isobars, Isotones and Isodiaphers

Type
Same
Different
Example / Point
Isotopes
Atomic number \((Z)\)
Mass number \((A)\)
\(_1H^1,*1H^2,*1H^3\)
Isobars
Mass number \((A)\)
Atomic number \((Z)\)
Different elements
Isotones
Number of neutrons \((A-Z)\)
Atomic and mass number
\(*1H^3\) and \(*2He^4\)
Isodiaphers
Difference between neutrons and protons
Atomic and mass number
\(*{92}U^{235}\) and \(*{90}Th^{231}\)
❖ Isotope Average Atomic Mass: \(M=\frac{n_1m_1+n_2m_2}{n_1+n_2}\)
▢ Forces Between Nucleons:

Table 1: Nuclear Forces

Force
Feature
Electrostatic Coulomb force
Exists between protons
Electrostatic effect
Causes repulsion between protons and nuclear instability
Nuclear force
Exists between any two nucleons
Strength
Strongest known force in nature
Charge dependence
Independent of charge
Range
Very short range, few fermi
Nature
Attractive at short range
Inverse square law
Not obeyed
Saturation
Saturable force
Dependence
Depends on spin / angular momentum of nuclei
Direction
Non-central force
❖ Stability Points:
  • •
    Nucleus is more stable when number of protons equals number of neutrons
  • •
    Even number of protons and even number of neutrons → more stable
  • •
    Magic numbers: 2, 8, 14, 20, 28, 50, 82, 126
▢ Mass Defect:
❖ Definition: Difference between sum of masses of constituent nucleons and actual mass of nucleus

Table 1: Mass Defect Formulae

Quantity
Formula
Mass defect
\(\Delta m=[Zm_p+(A-Z)m_n-M]\)
Packing fraction
\(f=\frac{\Delta m}{A}\)
Alternative packing fraction
\(f=\frac{M-A}{A}\)
❖ Note: Actual mass of nucleus is always less than sum of masses of constituent particles
▢ Binding Energy:
❖ Definition: Energy equivalent of mass defect of nucleus; energy binding the nucleus

Table 1: Binding Energy Formulae

Quantity
Formula
If \(\Delta m\) in amu
\(B.E.=\Delta m\times931MeV\)
If \(\Delta m\) in kg
\(B.E.=\Delta mc^2\)
Binding energy per nucleon
\(\frac{B.E.}{A}=\frac{\Delta m\times931}{A}\ MeV/nucleon\)
❖ Binding Energy Curve:
  • •
    Binding energy per nucleon rises sharply first
  • •
    Maximum value ≈ 8.8 MeV for iron \((Fe)\)
  • •
    After iron, curve falls slowly
  • •
    Medium nuclei are very stable
  • •
    Heavy nuclei are unstable and tend to split into medium nuclei
❖ Important Point: Binding energy per nucleon is a measure of nuclear stability
▢ Nuclear Fission:
❖ Definition: Splitting of a heavy nucleus into two or more lighter nuclei with release of large energy
❖ Condition: Usually heavy nuclei with \(A>230\)
❖ Discovered By: Hahn and Strassmann

Table 1: Uranium Fission

Fact
Answer
Common fission nucleus
\(*{92}U^{235}\)
Bombarding particle
Thermal neutron
Products
Barium + krypton + neutrons + energy
Reaction
\(*{92}U^{235}+_0n^1\rightarrow *{56}Ba^{141}+*{36}Kr^{92}+3_0n^1+Q\)
Mass defect
\(0.2153amu\)
Energy per fission
\(0.2153\times931=200.4MeV\)
Energy per fission in joule
\(3.2\times10^{-11}J\)
Natural uranium
0.7% \(U^{235}\), 99.3% \(U^{238}\)
❖ Conservation Laws:
  • •
    Linear momentum conserved
  • •
    Total energy conserved
  • •
    Charge conserved
  • •
    Number of nucleons conserved
❖ Reason: Fission occurs because binding energy per nucleon of medium nuclei is greater than that of heavy nuclei
▢ Chain Reaction:
❖ Definition: Continuous fission reaction maintained by secondary neutrons

Table 1: Chain Reaction

Term
Meaning
Secondary neutrons
Neutrons produced during fission causing further fission
Critical mass
Minimum mass of fissile material required for chain reaction
Uncontrolled chain reaction
Principle of atom bomb
Controlled chain reaction
Principle of nuclear reactor
▢ Nuclear Reactor:
❖ Definition: Device in which nuclear energy produced by controlled chain reaction is used for constructive purposes

Table 1: Nuclear Reactor Components

Component
Function
Examples
Nuclear fuel
Provides fissionable material
\(U^{235}, U^{236}, Pu^{239}, Pu^{240}, Pu^{241}\), thorium isotopes
Moderator
Slows down highly energetic neutrons
Heavy water, graphite, beryllium, paraffin
Control rods
Absorb excess neutrons and control chain reaction
Cadmium rods, boron rods
Neutron reflector
Prevents leakage of neutrons
Reflecting material
Coolant
Removes heat evolved in reactor core
Ordinary water, heavy water, \(CO_2\), liquid sodium, potassium, mercury
❖ Best Moderator: Heavy water
❖ Best Nuclear Fuel: \(Pu^{239}\)
▢ Neutron Types:

Table 1: Neutrons by Energy

Type
Energy
Slow neutrons
< 1 eV
Thermal neutrons
1 to 1.2 eV
Fast neutrons
> 1.2 eV
❖ Important Point: Neutrons are unaffected by electric field and are more effective than protons for nuclear reactions
▢ Nuclear Fusion:
❖ Definition: Fusion of two or more lighter nuclei to form a single heavier nucleus with release of energy

Table 1: Fusion Facts

Fact
Answer
Temperature required
\(10^7\) to \(10^8K\)
Also called
Thermonuclear reaction
Reason for high temperature
High kinetic energy needed to overcome electrostatic repulsion
Mass of product nucleus
Slightly less than sum of reacting nuclei
Energy released
\(E=\Delta mc^2\)
Mass converted into energy
About 1%
Compared with fission
Fusion releases more energy per unit mass
❖ Sample Reactions:
  • •
    \(_1H^1+_1H^1\rightarrow _1H^2+e^++\nu+0.42MeV\)
  • •
    \(_1H^2+_1H^2\rightarrow _2He^3+_0n^1+3.27MeV\)
  • •
    \(_1H^2+_1H^2\rightarrow _1H^3+_1H^1+4.03MeV\)
❖ Hydrogen Bomb: Based on nuclear fusion
▢ Stellar Energy:
❖ Source: Solar / stellar energy is due to fusion of hydrogen into helium
❖ Star Matter: Hydrogen and helium together form about 99% by weight of star matter

Table 1: Thermonuclear Cycles

Cycle
Main Condition
Net Result
Proton-proton cycle
Dominant in stars with temperature less than sun
Four hydrogen nuclei combine to form helium
Carbon-nitrogen cycle
Dominant in hotter stars
Carbon acts as catalyst; hydrogen converts to helium
❖ Proton-Proton Cycle:
  1. \(_1H^1+_1H^1\rightarrow _1H^2+e^++\nu+Q_1\)
  2. \(_1H^1+_1H^2\rightarrow _2He^3+\gamma+Q_2\)
  3. \(_2He^3+_2He^3\rightarrow _2He^4+2_1H^1+Q_3\)
  4. Net: \(4_1H^1\rightarrow _2He^4+2e^++Q\)
❖ Carbon-Nitrogen Cycle:
  1. \(_6C^{12}+_1H^1\rightarrow _7N^{13}+\gamma+Q_1\)
  2. \(_7N^{13}\rightarrow _6C^{13}+e^++\nu+Q_2\)
  3. \(_6C^{13}+_1H^1\rightarrow _7N^{14}+\gamma+Q_3\)
  4. \(_7N^{14}+_1H^1\rightarrow _8O^{15}+\gamma+Q_4\)
  5. \(_8O^{15}\rightarrow _7N^{15}+e^++\nu+Q_5\)
  6. \(_7N^{15}+_1H^1\rightarrow _6C^{12}+_2He^4+Q_6\)
  7. Net: \(4_1H^1\rightarrow _2He^4+2e^++Q\)
▢ Particle Antiparticle Pairs:

Table 1: Particle-Antiparticle Pairs

Particle
Antiparticle
Electron
Positron
Proton
Antiproton
Neutron
Antineutron
Neutrino
Antineutrino
❖ Pair Production: Minimum gamma photon energy for electron-positron pair production is \(1.02MeV\)
❖ Annihilation: When particle and antiparticle annihilate, mass converts into energy
▢ Quark Combination:

Table 1: Quark Composition

Combination
Particle
Charge
\(uud\)
Proton
+1
\(udd\)
Neutron
0
\(\bar u\bar u\bar d\)
Antiproton
-1
\(\bar u\bar d\bar d\)
Antineutron
0
\(u\bar u\)
\(\pi^0\) meson
0
\(u\bar d\)
\(\pi^+\) meson
+1
\(\bar ud\)
\(\pi^-\) meson
-1
▢ Nuclear Decay:

Table 1: Decay Modes

Decay
Equation
Change
\(\alpha\)-decay
\(_Z^AX\rightarrow _{Z-2}^{A-4}Y+_2He^4+Q\)
\(A\downarrow4, Z\downarrow2\)
\(\beta^-\)-decay
\(_Z^AX\rightarrow *{Z+1}^{A}Y+*{-1}e^0+\bar\nu+Q\)
\(A\) same, \(Z\uparrow1\)
\(\beta^+\)-decay
\(_Z^AX\rightarrow *{Z-1}^{A}Y+*{+1}e^0+\nu+Q\)
\(A\) same, \(Z\downarrow1\)
\(\gamma\)-decay
\(_Z^AX^*\rightarrow _Z^AX+\gamma\)
\(A,Z\) same; energy decreases
K-capture
\(*Z^AX+*{-1}e^0\rightarrow _{Z-1}^{A}Y+X-ray+\nu\)
Nucleus captures orbital electron
❖ Neutron Decay: \(_0n^1\rightarrow *1H^1+*{-1}e^0+\bar\nu\)
❖ Proton Conversion in Positron Decay: \(p\rightarrow n+e^++\nu\)
▢ Isoelectronic, Isoster and Isolobel:

Table 1: Extra Terms

Term
Meaning
Example
Isoelectronic
Same number of electrons
\(CO, CN^-\)
Isoster
Both isoelectronic and isoatomic
\(N_2O, CO_2\)
Isolobel
Same geometry and hybridization
\(SO_2, SnCl_2\)
▢ Read and Digest:

Table 1: Important Points

Fact
Answer
Nucleus discovered by
Rutherford
Proton discovered by
Goldstein
Neutron discovered by
Chadwick
Nuclear holocaust
Large-scale destruction due to nuclear weapons
1 amu
931 MeV
Heavy water
Best moderator
Solar energy source
Nuclear fusion
Hydrogen bomb
Based on nuclear fusion
Atom bomb
Based on uncontrolled nuclear fission
Nuclear reactor
Based on controlled nuclear fission
Critical mass
Minimum mass needed for chain reaction
Average binding energy per nucleon
About 8 MeV
Maximum binding energy per nucleon
For iron
Nuclear density order
\(10^{17}kgm^{-3}\)
Binding energy
Measure of nuclear stability
Cosmic rays
Mainly protons at top of atmosphere
Meson mass
Nearly \(0.013amu\)
Fermions
Half-integral spin particles constituting matter
Neutrino
No charge, spin present, very small/no mass
Positron
Antiparticle of electron with same mass and opposite charge
Fusion condition
High temperature and high pressure
Fission product mass
Less than parent nucleus mass
In fusion
Binding energy per nucleon increases
▢ Objective Answer Key:

Table 1: Nuclear Physics MCQ Answers

Q
Ans
1
c
2
a
3
a
4
c
5
a
6
a
7
b
8
b
9
a
10
a
11
c
12
b
13
b
14
c
15
d
16
d
17
c
18
b
19
b
20
c
21
d
22
b
23
c
24
d
25
a
26
b
27
d
28
a
29
c
30
d
31
b
32
b
33
c
34
d
35
a
36
b
37
b
38
d
39
a
40
c
41
a
42
b
43
c
44
d
45
b
46
b
47
c
48
d
49
a
50
c
51
c
52
c
53
a
54
c
55
a
56
b
57
b
▢ High-Yield Recall:

Table 1: Nuclear Physics One-Liners

Fact
Answer
Nucleons
Protons + neutrons
Atomic number
\(Z\)
Mass number
\(A\)
Neutrons
\(A-Z\)
Nuclear radius
\(R=R_0A^{1/3}\)
Nuclear density
\(\rho=\frac{3m}{4\pi R_0^3}\)
1 amu
\(931MeV\)
Isotopes
Same \(Z\), different \(A\)
Isobars
Same \(A\), different \(Z\)
Isotones
Same neutrons
Isodiaphers
Same \(N-Z\)
Mass defect
\(\Delta m=Zm_p+(A-Z)m_n-M\)
Binding energy
\(B.E.=\Delta m\times931MeV\)
BE per nucleon
\(\frac{\Delta m\times931}{A}\)
Most stable nucleus region
Iron region
Fission
Heavy nucleus → lighter nuclei
Fusion
Light nuclei → heavier nucleus
Fission energy per \(U^{235}\)
About 200 MeV
Critical mass
Minimum mass for chain reaction
Moderator
Slows down neutrons
Control rod
Absorbs excess neutrons
Best moderator
Heavy water
Best nuclear fuel
\(Pu^{239}\)
Nuclear reactor
Controlled fission
Atom bomb
Uncontrolled fission
Hydrogen bomb
Fusion
Solar energy
Fusion of hydrogen into helium
Pair production threshold
\(1.02MeV\)
Proton quark combination
\(uud\)
Neutron quark combination
\(udd\)
Alpha decay
\(A-4,Z-2\)
Beta minus decay
\(A\ same,Z+1\)
Beta plus decay
\(A\ same,Z-1\)
Gamma decay
\(A,Z\ same\)
K-capture
Orbital electron captured by nucleus
Q1.
When a particles and its antiparticle are annihilated the energy released is E what is the mass of each particle?
📅MOE 2012 & 2013
Q2.
Which of the following is correct representation of a nuclear reaction (notation carry usual meanings)?
📅MOE 2068
Q3.
Correct decay scheme for an emission of β- emission from a radioactive nuclear AZX may be
📅MOE 2010
Q4.
A nucleus AZX decays to AZ-1Y plus an additional nuclear particle. The resulting particle may be
📅MOE 2009
Q5.
The fusion of hydrogen into helium is more likely to take place:
📅KU 2011/2017
Q6.
Energy is radiated in which of the following?
📅KU 2010
Q7.
Positron has
📅KU 2010
Q8.
The quark combination of proton is
📅KU 2010
Q9.
Which is emitted in β-emission?
📅BP 2012
Q10.
The average distance between nucleus & electron is
📅BP 2012
Q11.
Which of the following has similar mass to electron?
📅T.E. 2009
Q12.
The binding energy of hydrogen atom is
📅I.E. 2009
Q13.
The nuclear radius of nucleus with nucleon number 16 is 3×10-15 m. With nucleon number 128, radius is
📅I.E. 2009
Q14.
In fusion, % of mass converted into energy is about
📅I.E. 2009
Q15.
n = p + e + X, where n = neutron, p is proton, e is electron. What is X?
📅I.E. 2013
Q16.
Find out the no. of neutrons released in the fission of 235U into 144Xe and 90Sr.
📅I.E. 2010
Q17.
The neutron possesses
📅I.E. 2010
Q18.
The energy released in the fission of one nucleus is 200 MeV. The number of fissions per second required to produce 3.2 W of power is
📅I.E. 2010
Q19.
Following revolves around the nucleus.
📅I.E. 2012
Q20.
Nucleon is
📅I.E. 2012
Q21.
In nuclear reactor, cadmium rod is used as
📅BP 2014
Q22.
The packing fraction for 14N isotope, whose mass is 14.003 amu is
📅BP 2014
Q23.
The binding energies per nucleon for a deuteron and an α-particle are x1 and x2 respectively. What will be energy Q released in the reaction: (1H2 + 1H2 → 2He4 + Q)
📅BP 2014
Q24.
The mass defect for the nucleus of helium is 0.0303 amu. What is the binding energy per nucleon for helium in MeV?
📅BP 2014
Q25.
A nucleus of 210Po originally at rest emits an α-particle with speed v. What will be recoil speed of the daughter nucleus?
📅BP 2014
Q26.
Energy released in the fission of a single 235U nucleus is 200 MeV. The fission rate of a 235U-filled reactor operating at a power level of 5 W is
📅BP 2014
Q27.
The nuclear radius of a nucleus with nucleon number 16 is 3 × 10-15 m. With nucleon number 128, radius would be
📅BP 2014
Q28.
The ratio of the radii of the nuclei 27Al and 125Te is approximately
📅BP 2014
Q29.
If the mass defect in the formation of helium from hydrogen is 0.5%, then the energy obtained in kWh, in forming helium from 1 kg of hydrogen will be
📅BP 2014
Q30.
The binding energies per nucleon of 7Li and 4He are 5.6 MeV and 7.06 MeV respectively, then the energy of the reaction 7Li + p → 2[4He] will be
📅BP 2014
Q31.
The binding energy of two nuclei P and Q are x joule and y joule respectively. If 2x > y, then the energy released in the reaction P + P → Q will be
📅BP 2014
Q32.
A nucleus ruptures into two nuclear parts which have their velocity ratio equal to 2:1. What will be the ratio of their nuclear radius?
📅BP 2014
Q33.
The binding energy per nucleon of deuteron (2H) and helium (4He) is 1.1 MeV and 7 MeV respectively. If two deuterons fuse to form a single helium atom, then the energy released is
📅BP 2014
Q34.
Two deuterons each of mass m fuse to form helium resulting in release of energy E. The mass of helium formed is
📅BP 2014
Q35.
The nuclei of which one of the following pairs of nuclei are isotones?
📅BP 2014
Q36.
Consider the nuclear reaction X200 → A100 + B100 + Energy. If the binding energy per nucleon for X, A and B is 7.4 MeV, 8.2 MeV and 8.2 MeV respectively, what is the energy released?
📅BP 2014
Q37.
A gamma ray photon creates an electron-positron pair. If the rest mass of electron is 0.51 MeV and the total kinetic energy of electron-positron pair is 0.78 MeV, the energy of gamma ray photon must be:
📅BP 2014
Q38.
To produce proton-antiproton pair, the minimum energy of γ-rays photon must be
📅BP 2014
Q39.
The minimum energy needed to break C12 into three α-particles: mass of C12 = 12.000 and mass of α-particle = 4.0038
📅BP 2014
Q40.
Two nucleons are at a separation of 1 fm. The net force between them is F1 if both are neutrons, F2 if both are protons and F3 if one is proton and the other is a neutron.
📅BP 2014
Q41.
In the nucleus of 23Na11, the number of protons, neutrons and electrons is
📅BP 2014
Q42.
In the nuclear reaction 10B + 4He → 13N + X, the missing term is a
📅BP 2014
Q43.
In the nuclear reaction 12C → 12B + β+ + X, what does X stand for?
📅BP 2014
Q44.
A nucleus AXZ emits one alpha particle and two β-particles. The resulting nucleus is
📅BP 2014
Q45.
The mass of a proton is 1.007825 u and mass of neutron is 1.008665 u, then the binding energy per nucleon is 35Cl if its mass is 34.980 u is:
📅BP 2014
Q46.
A nucleus with mass number 200 initially at rest emits an alpha particle. If the Q of the reaction is 5.5 MeV, Calculate the K.E. of the α-particle
📅BP 2014
Q47.
The radius of Al nucleus, in fermi, is nearly
📅MOE Curriculum
Q48.
The approx radius of nucleus of Cu in fermi is
📅MOE 2008
Q49.
The radius of gold nucleus is approximately
📅MOE 2065
Q50.
A positron has the same mass as
📅MOE 2063
Q51.
Radius of Al (At wt = 27) nucleus in fermi is
📅MOE 2056
Q52.
Binding energy per nucleon is highest for the atom
📅T.E. 2006
Q53.
Neutrino is a particle which has
📅T.E. 2004
Q54.
When fuel of the star exhausts
📅BPKIHS 2002
Q55.
According to Standard Model, spin half particles are known as
📅KU 2015
Q56.
The sun radiates energy in all directions. The average radiation received on the earth's surface from the sun per second is 1.4 kW/m². The average sun-earth distance is 1.5×1011 m. The mass lost by the sun per day is
📅KU 2015
Q57.
The source of energy of sun is
📅KU 2016