Nuclei — NEET notes

Nuclear Composition

Every atom's identity is locked inside a tiny nucleus — just 10⁻¹⁵ m across — packed with protons and neutrons. The number of protons (Z) decides WHICH element it is; the total nucleons (A) decides WHICH isotope. Change Z, you change the element. Change A, you just get a heavier or lighter version of the same element.

R = R_0 A^ 1/3

Atomic Number & Mass Number

Z (protons) identifies the element. A (protons + neutrons) identifies the specific nuclide. These two numbers tell you everything about nuclear composition.

The atomic number Z = number of protons in the nucleus. The mass number A = total number of nucleons (protons + neutrons). Number of neutrons N = A - Z .

A = Z + N

A nuclide is represented as ^A_Z X , where X is the chemical symbol. For example, ^ 12 _6C means carbon with 6 protons and 6 neutrons (A = 12).

• Z (atomic number) = number of protons = number of electrons in a neutral atom
• A (mass number) = number of protons + neutrons = total nucleons
• N (neutron number) = A - Z
• Notation: ^A_Z X — mass number on top, atomic number on bottom
• Protons and neutrons are collectively called nucleons

Isotopes, Isobars & Isotones

These three terms classify nuclides based on which nuclear number they share. NEET loves testing whether you can tell them apart.

• Isotopes have identical chemical properties (same electrons) but different nuclear properties
• Isobars have different chemical properties (different Z = different element)
• Hydrogen has three isotopes: protium ( ^1 H), deuterium ( ^2 H), tritium ( ^3 H)
• Carbon-12 and Carbon-14 are isotopes — C-14 is radioactive and used in carbon dating

Nuclear Size

The nucleus is about 10,000 times smaller than the atom. Nuclear radius follows a simple formula that shows volume is proportional to the number of nucleons — like stacking identical marbles.

R = R_0 A^ 1/3

Here R_0 1.2 10^ -15 m (1.2 fm). This means the nuclear volume V = 4 3 R^3 A . Since volume is proportional to mass number, the nuclear density is constant for all nuclei.

_ nuclear 2.3 10^ 17 kg/m^3

• R_0 1.2 fm = 1.2 10^ -15 m (1 fermi = 10^ -15 m)
• Nuclear volume V A — each nucleon occupies roughly the same volume
• Nuclear density 2.3 10^ 17 kg/m³ — same for ALL nuclei (hydrogen to uranium)
• This density is 10^ 14 times denser than water — a teaspoon of nuclear matter would weigh billions of tonnes
• Atom radius 10^ -10 m vs nucleus radius 10^ -15 m → nucleus is 100,000 times smaller

Quick recall

If the mass number of a nucleus becomes 8 times larger, by what factor does the radius increase?

R A^ 1/3 , so if A 8A , then R (8)^ 1/3 R = 2R . The radius doubles.

Nuclear Mass & Mass Defect

Here's the mind-bending truth: a nucleus weighs LESS than the sum of its parts. The 'missing' mass (mass defect) has been converted into binding energy via E = mc² — the energy that glues the nucleus together. This is the most direct proof of Einstein's famous equation.

BE = m c^2

Atomic Mass Unit

Nuclear masses are tiny — we need a special unit. One atomic mass unit (u) is defined as 1/12th the mass of a Carbon-12 atom, and it connects directly to energy via E = mc².

Measuring nuclear masses in kilograms gives absurdly small numbers ( 10^ -27 kg). The atomic mass unit (u) is a much more practical scale. By definition, 1 u = 1 12 the mass of a ^ 12 C atom.

1 u = 1.660539 10^ -27 kg

Using Einstein's mass-energy equivalence E = mc^2 , we can find the energy equivalent of 1 u:

1 u c^2 = 931.5 MeV

• 1 u = 1.660539 10^ -27 kg (memorize this)
• Energy equivalent: 1 u c^2 = 931.5 MeV (memorize this!)
• Proton mass m_p = 1.00728 u = 938.3 MeV/ c^2
• Neutron mass m_n = 1.00866 u = 939.6 MeV/ c^2
• Electron mass m_e = 0.00055 u = 0.511 MeV/ c^2
• Neutron is slightly heavier than proton — this is why free neutrons are unstable (beta decay)

Mass Defect

When nucleons bind together, some mass disappears. This mass defect is not lost — it has been converted into binding energy that holds the nucleus together.

For a nucleus ^A_Z X with Z protons and N = A - Z neutrons, the mass defect is the difference between the total mass of free nucleons and the actual nuclear mass:

m = [Z m_p + (A - Z) m_n] - M_ nucleus

Since NCERT tables give atomic masses (including electrons), we use an equivalent formula with atomic masses. The electron masses cancel out:

m = [Z m(^1H) + (A - Z) m_n] - M(^A_Z X)

• m is ALWAYS positive for stable nuclei — the bound system weighs less
• Use atomic masses (not nuclear masses) — electron masses cancel automatically
• m(^1H) = 1.007825 u (hydrogen atom mass, includes one electron)
• The mass defect is typically 0.001 to 0.009 u per nucleon

Binding Energy

The binding energy is the energy equivalent of the mass defect. It represents the energy needed to break the nucleus apart into free nucleons — or equivalently, the energy released when free nucleons come together to form the nucleus.

BE = m c^2

BE = m 931.5 MeV (when m is in u)

The binding energy has two equivalent interpretations: (1) the energy required to disassemble the nucleus into free nucleons, and (2) the energy released when free nucleons assemble into the nucleus.

• BE = m 931.5 MeV (quick conversion when m in u)
• Higher BE = more stable nucleus (harder to break apart)
• BE of ^4He = 28.3 MeV, ^ 56 Fe = 492 MeV, ^ 235 U = 1784 MeV
• BE is always positive for stable nuclei
• 1 MeV = 10^6 eV = 1.6 10^ -13 J

Binding Energy Per Nucleon

Total binding energy tells you how strong a nucleus is overall, but binding energy per nucleon (BE/A) tells you how stable it is per particle — and THAT is the key to understanding why some nuclei fuse, some fission, and iron sits at the top as the most stable nucleus in the universe.

BE A = m 931.5 A MeV/nucleon

The BE/A Curve

Plot BE/A vs A for all nuclei and you get the famous binding energy curve. It peaks at iron (A ≈ 56), explaining why energy is released when light nuclei fuse AND when heavy nuclei split.

The binding energy per nucleon (BE/A) is the average energy needed to remove one nucleon from the nucleus. A higher BE/A means each nucleon is bound more tightly — the nucleus is more stable.

• BE/A is LOW for very light nuclei ( ^2H : 1.1 MeV) — weakly bound
• BE/A RISES steeply through ^4He (7.07 MeV), ^ 12 C (7.68 MeV), ^ 16 O (7.98 MeV)
• BE/A PEAKS at ^ 56 Fe with 8.75 MeV/nucleon — the most stable nucleus
• BE/A slowly DECREASES for heavy nuclei: ^ 238 U has about 7.6 MeV/nucleon
• Most nuclei have BE/A between 7.5 and 8.8 MeV — remarkably constant

Why Fusion & Fission Release Energy

Any nuclear reaction that moves nuclei TOWARD the iron peak on the BE/A curve releases energy. Light nuclei move right (fusion). Heavy nuclei move left (fission). Iron can do neither — it's already at the top.

Fusion (light nuclei combine): When two light nuclei fuse, the product has a HIGHER BE/A than the reactants. The increase in binding energy is released as kinetic energy. Example: ^2H + ^2H ^4He — BE/A jumps from ~1.1 to 7.07 MeV.

Fission (heavy nucleus splits): When a heavy nucleus splits, the fragments have a HIGHER BE/A than the parent. The increase in binding energy is released. Example: ^ 235 U (BE/A ≈ 7.6) splits into fragments with BE/A ≈ 8.5.

• Energy released = (total BE of products) − (total BE of reactants)
• If products have HIGHER total BE → energy is RELEASED (exothermic)
• If products have LOWER total BE → energy must be SUPPLIED (endothermic)
• Fusion of light nuclei ( A < 56 ): products move RIGHT on curve → higher BE/A → energy released
• Fission of heavy nuclei ( A > 56 ): products move LEFT on curve → higher BE/A → energy released
• Iron ( A = 56 ): already at peak → neither fusion nor fission releases energy

Quick recall

Why can't we extract energy by fusing two iron nuclei?

Iron-56 is at the PEAK of the BE/A curve. Fusing two iron nuclei would create a heavier nucleus with LOWER BE/A — this would require energy INPUT, not release it. Iron is the endpoint of both fusion and fission.

Special Peaks & Magic Numbers

Certain nuclei show unusually high BE/A — appearing as sharp peaks on the curve. These correspond to 'magic numbers' of protons or neutrons, analogous to noble gas electron configurations.

• Magic numbers: 2, 8, 20, 28, 50, 82, 126
• ^4He (Z=2, N=2) — doubly magic → BE/A = 7.07, unusually high for A = 4
• ^ 16 O (Z=8, N=8) — doubly magic → BE/A = 7.98
• ^ 40 Ca (Z=20, N=20) — doubly magic
• ^ 208 Pb (Z=82, N=126) — doubly magic → most stable heavy nucleus
• Even-even nuclei (even Z, even N) are more stable than odd-odd nuclei

Nuclei with even numbers of both protons and neutrons are more abundant in nature — they have more stable configurations. There are only 4 stable odd-odd nuclei: ^2H , ^6Li , ^ 10 B , ^ 14 N .

Nuclear Forces

What holds the nucleus together? Protons repel each other electrically, yet nuclei are incredibly stable. The answer is the strong nuclear force — the strongest force in nature, but with an incredibly short range (~few femtometres). It's like super-glue that only works when things are touching.

Strong force range few fm (10^ -15 m)

Properties of Nuclear Force

The strong nuclear force has unique properties that distinguish it from electromagnetic and gravitational forces. Understanding these properties explains why nuclei exist at all.

• Strongest force in nature — about 100 times stronger than electromagnetic force at nuclear distances
• Very short range — effective only up to few femtometres (fm). Beyond 3 fm, it drops to practically zero
• Charge independent — force between p-p = p-n = n-n (all equal at the same distance)
• NOT a central force — depends on spin and other quantum properties, not just distance
• Saturated — each nucleon interacts only with its nearest neighbors, not with ALL other nucleons
• Attractive at medium range ( > 0.8 fm) but repulsive at very short range ( < 0.8 fm) — prevents nucleons from collapsing

The attractive-repulsive nature is crucial: at distances > 0.8 fm, the force is strongly attractive (pulling nucleons together). Below 0.8 fm, it becomes repulsive (preventing collapse). This is why nuclear density is constant — nucleons maintain a fixed average spacing.

Comparing Forces

Nature has four fundamental forces. Inside the nucleus, the strong force dominates. At atomic scales, electromagnetic wins. At cosmic scales, gravity rules. Each has its domain.

Inside the nucleus ( r 1 fm): strong force electromagnetic repulsion gravity. The strong force wins by a factor of ~100 over electromagnetic. But beyond ~3 fm, the strong force vanishes and only electromagnetic and gravitational remain.

• At r 1 fm: F_ strong 100 F_ EM
• At r 10 fm: F_ strong 0 (out of range)
• Gravity is 10^ -36 times weaker than the strong force — completely negligible inside nuclei
• For heavy nuclei ( Z > 82 ): Coulomb repulsion between many protons overwhelms the short-range strong force → nucleus becomes unstable → radioactivity

Radioactive Decay

Unstable nuclei shed excess energy by emitting particles or radiation — alpha (helium nuclei), beta (electrons/positrons), or gamma (pure energy). Each decay type changes the nucleus differently, and conservation laws tell you exactly what comes out. This is nature's way of moving nuclei toward stability.

^A_Z X ^ A-4 _ Z-2 Y + ^4_2 He

Alpha Decay

In alpha decay, a heavy unstable nucleus emits an alpha particle (²He⁴ = 2 protons + 2 neutrons), reducing both its mass number by 4 and atomic number by 2. This is how heavy nuclei lose bulk to become more stable.

^A_Z X ^ A-4 _ Z-2 Y + ^4_2 He ( particle)

Example: Uranium-238 alpha decay: ^ 238 _ 92 U ^ 234 _ 90 Th + ^4_2He . The parent (U-238) loses 4 nucleons and 2 protons, becoming thorium-234.

• A decreases by 4, Z decreases by 2 → element moves 2 places LEFT in periodic table
• Alpha particles are ^4_2He nuclei — charge +2e, mass 4u
• Typical alpha energies: 4–9 MeV
• Alpha particles have HIGH ionizing power but LOW penetrating power (stopped by paper/skin)
• Common in heavy nuclei ( Z > 82 ) where Coulomb repulsion dominates
• The alpha particle must quantum tunnel through the Coulomb barrier to escape

Quick recall

After 3 alpha decays and 2 beta-minus decays, how do A and Z change?

3 alpha decays: A decreases by 3 4 = 12 , Z decreases by 3 2 = 6 . 2 beta-minus decays: A unchanged, Z increases by 2. Net: A decreases by 12, Z decreases by 4.

Beta Decay

In beta decay, a neutron converts to a proton (β⁻) or a proton converts to a neutron (β⁺). The mass number stays the same, but the element changes. An elusive neutrino carries away the missing energy.

Beta-minus ( ^- ) decay: A neutron converts into a proton, emitting an electron and an antineutrino.

^A_Z X ^A_ Z+1 Y + e^- + _e

Beta-plus ( ^+ ) decay: A proton converts into a neutron, emitting a positron and a neutrino.

^A_Z X ^A_ Z-1 Y + e^+ + _e

• ^- : A unchanged, Z increases by 1 → element moves 1 place RIGHT in periodic table
• ^+ : A unchanged, Z decreases by 1 → element moves 1 place LEFT
• The emitted electron/positron is NOT from an orbital — it's created during the decay
• Neutrino ( ) / antineutrino ( ): nearly massless, neutral, very weakly interacting
• Beta energy spectrum is continuous (0 to KE_ ) — Pauli postulated the neutrino to explain this
• Underlying process: n p + e^- + _e (mediated by the weak nuclear force)

Gamma Decay

Gamma decay is the nuclear equivalent of an excited atom emitting a photon. The nucleus drops from a higher energy state to a lower one, emitting a high-energy gamma ray. No change in A or Z.

^A_Z X^* ^A_Z X +

The asterisk (*) denotes an excited nuclear state. After alpha or beta decay, the daughter nucleus is often left in an excited state — it then emits one or more gamma rays to reach the ground state.

• No change in A or Z — the nucleus stays the same element
• Gamma rays are high-energy photons: E = h , typically 0.1–10 MeV
• Gamma rays have the HIGHEST penetrating power (need thick lead/concrete to stop them)
• Gamma rays have the LOWEST ionizing power among , ,
• Gamma emission usually FOLLOWS alpha or beta decay (daughter is often in excited state)

Conservation Laws in Decay

Every nuclear decay must conserve charge (Z), mass number (A), energy, and momentum. These conservation laws let you predict the products of any decay reaction.

In any nuclear reaction or decay, the following quantities are always conserved:

• Mass number (A): Total nucleons on left = total nucleons on right
• Atomic number (Z): Total charge on left = total charge on right
• Energy: Total energy (rest mass + kinetic) is conserved. The Q-value gives the net energy released
• Linear momentum: Total momentum before = total momentum after
• Lepton number: Number of leptons is conserved (explains why neutrinos appear in beta decay)

The Q-value of a decay is the energy released: Q = (M_ parent - M_ products ) 931.5 MeV. If Q > 0 , the decay is energetically possible (exothermic). If Q < 0 , it cannot occur spontaneously.

Q = (m_ parent - m_ products ) c^2

Quick recall

^ 226 _ 88 Ra ^A_ZRn + ^4_2He . Find A and Z of Rn.

Balance A: 226 = A + 4 A = 222 . Balance Z: 88 = Z + 2 Z = 86 . The product is ^ 222 _ 86 Rn (radon-222).

Radioactive Decay Law

Radioactive decay is random for any single atom but perfectly predictable for large numbers. The number of undecayed atoms drops exponentially — N = N₀e^(-λt). Half-life is the time for half the atoms to decay, and activity measures how 'radioactive' a sample is right now. These are NEET's favorite numerical topics in nuclear physics.

N = N_0 e^ - t

Exponential Decay Law

The rate of decay is proportional to the number of atoms present. This gives the famous exponential decay equation — the same math as RC discharge in circuits!

The fundamental law: the rate of disintegration is proportional to the number of undecayed nuclei at that instant.

dN dt = - N

Solving this differential equation with initial condition N(0) = N_0 gives:

N(t) = N_0 e^ - t

• N_0 = initial number of radioactive nuclei at t = 0
• N(t) = number of undecayed nuclei remaining at time t
• = decay constant (probability of decay per unit time, unit: s ^ -1 )
• Larger → faster decay → shorter half-life
• The negative sign indicates N is DECREASING with time
• The number of nuclei that have DECAYED = N_0 - N = N_0(1 - e^ - t )

Half-Life

Half-life T₁/₂ is the most intuitive measure of decay rate. After one half-life, half the atoms remain. After two, one-quarter. After n half-lives, 1/2ⁿ remain. Simple and powerful.

At t = T_ 1/2 , exactly half the original nuclei remain: N = N_0/2 . Substituting into the decay law:

T_ 1/2 = 2 = 0.693

After n half-lives, the fraction remaining is:

N = N_0 2^n where n = t T_ 1/2

• T_ 1/2 = 0.693/ — memorize this relationship
• Half-life is a CONSTANT — it does not change with time or the amount of substance
• Half-life does NOT mean 'all atoms decay in 2T_ 1/2 ' — it's exponential, not linear
• Range of half-lives in nature: 10^ -22 s (very unstable) to 10^ 10 years (nearly stable)
• ^ 238 U : T_ 1/2 = 4.47 10^9 years (age of Earth!). ^ 14 C : T_ 1/2 = 5730 years (carbon dating)

Quick recall

A radioactive sample has a half-life of 2 hours. What fraction remains after 8 hours?

n = t/T_ 1/2 = 8/2 = 4 half-lives. Fraction remaining = 1/2^4 = 1/16 . So 1/16th of the original sample remains.

Mean Life

Mean life (τ) is the average time a radioactive atom survives before decaying. It's slightly longer than the half-life by a factor of 1/ln2 ≈ 1.44.

= 1 = T_ 1/2 0.693 = 1.44 T_ 1/2

Mean life is the time at which the number of nuclei drops to 1/e ( 37 % ) of the original value. Compare with half-life where it drops to 50%.

• = 1/ — reciprocal of decay constant
• = 1.44 T_ 1/2 — mean life is always longer than half-life
• At t = : N = N_0/e 0.37 N_0 (37% remaining)
• At t = T_ 1/2 : N = N_0/2 = 0.50 N_0 (50% remaining)
• Total number of decays from t = 0 to equals N_0 (every atom eventually decays)

Activity

Activity (A or R) measures the rate of disintegration — the number of decays per second. It's what a Geiger counter measures. Activity also decreases exponentially with the same half-life.

A = N = N_0 e^ - t = A_0 e^ - t

Activity is the magnitude of dN/dt — the number of disintegrations per second. It's directly proportional to N , so it decays exponentially with the same time constant.

• A = N = |dN/dt| — activity equals decay constant times number of atoms
• SI unit: becquerel (Bq) — 1 Bq = 1 disintegration per second
• Old unit: curie (Ci) — 1 Ci = 3.7 10^ 10 Bq (activity of 1 g of Ra-226)
• Activity has the SAME half-life as the number of atoms: A = A_0/2^n
• Higher activity = more dangerous (more radiation emitted per second)
• As N decreases, activity decreases proportionally

Quick recall

A sample has an initial activity of 800 Bq. After 3 half-lives, what is the activity?

A = A_0/2^n = 800/2^3 = 800/8 = 100 Bq.

Nuclear Energy

Nuclear energy comes in two flavors: fission (splitting heavy nuclei like uranium) and fusion (joining light nuclei like hydrogen). Fission powers nuclear reactors and atom bombs. Fusion powers the Sun and hydrogen bombs. Both release millions of times more energy per kilogram than any chemical reaction — because they convert mass directly into energy via E = mc².

E = m c^2

Nuclear Fission

When a heavy nucleus (like U-235) absorbs a neutron, it becomes unstable and splits into two medium-mass fragments, releasing ~200 MeV of energy and 2-3 more neutrons. This is the basis of nuclear power and atomic bombs.

Fission was discovered by Otto Hahn and Fritz Strassmann (1938) and explained by Lise Meitner. When ^ 235 U absorbs a slow (thermal) neutron, it forms ^ 236 U^* which is highly unstable and splits:

^ 235 _ 92 U + ^1_0 n ^ 236 _ 92 U^* ^ 141 _ 56 Ba + ^ 92 _ 36 Kr + 3 ^1_0 n + Q

The energy released per fission event is approximately 200 MeV — about 10^7 times the energy from burning one molecule of TNT.

• Fissile materials: ^ 235 U (natural, 0.7% abundance), ^ 239 Pu (man-made)
• Each fission releases 2–3 neutrons plus ~200 MeV energy
• Products are NOT unique — fission fragments vary statistically (many possible fragment pairs)
• Energy appears as: KE of fragments (~165 MeV), KE of neutrons (~5 MeV), gamma rays (~15 MeV), beta decay energy (~15 MeV)
• Slow (thermal) neutrons are needed to fission U-235 — fast neutrons are less effective
• Natural uranium is mostly ^ 238 U (99.3%) which does NOT fission with slow neutrons

Chain Reaction

Each fission releases 2-3 neutrons which can trigger more fissions, creating a self-sustaining chain reaction. Control this chain and you get a nuclear reactor. Let it run away and you get a bomb.

The neutron multiplication factor k determines the fate of the chain reaction:

• k < 1 (subcritical): reaction dies out — fewer neutrons each generation
• k = 1 (critical): reaction is sustained at constant rate — this is what a nuclear reactor maintains
• k > 1 (supercritical): reaction grows exponentially — this is what happens in a bomb
• Critical mass: minimum mass of fissile material needed to sustain a chain reaction
• For U-235: critical mass 15 kg (a sphere about the size of a grapefruit)

Not all neutrons from fission cause further fission. Some escape the material, some are absorbed by non-fissile atoms. The balance between production and loss determines k .

Nuclear Reactor

A nuclear reactor maintains a controlled chain reaction (k = 1) to produce steady energy. The key components work together to slow neutrons, sustain the reaction, and extract heat safely.

A nuclear reactor has five essential components that work together:

• Fuel: Enriched uranium ( ^ 235 U , 2-4% enrichment) or plutonium-239
• Moderator: Slows fast neutrons to thermal speeds. Materials: heavy water ( D_2O ), graphite, or light water. Slow neutrons have higher probability of causing fission in U-235
• Control rods: Absorb excess neutrons to maintain k = 1 . Materials: cadmium (Cd) or boron. Push in → absorb more neutrons → reduce power. Pull out → absorb fewer → increase power
• Coolant: Carries heat from reactor core to steam generator. Materials: water, heavy water, or liquid sodium
• Shielding: Thick concrete + steel walls to block radiation from escaping

Nuclear Fusion

Fusion combines light nuclei (like hydrogen isotopes) into heavier ones, releasing even more energy per nucleon than fission. It powers every star in the universe, but achieving controlled fusion on Earth remains one of science's greatest challenges.

The proton-proton chain (simplified) — the fusion reaction that powers the Sun:

4 ^1_1H ^4_2He + 2e^+ + 2 _e + 26.7 MeV

Thermonuclear fusion on Earth (the reaction most promising for reactors):

^2_1H + ^3_1H ^4_2He + ^1_0 n + 17.6 MeV

• Requires extreme temperatures ( 10^7 to 10^8 K) to overcome Coulomb repulsion between positively charged nuclei
• At these temperatures, matter exists as plasma (ionized gas)
• Energy per nucleon from fusion > energy per nucleon from fission (steeper BE/A curve on left side)
• Fusion produces NO long-lived radioactive waste (unlike fission)
• Hydrogen bomb = uncontrolled fusion triggered by a fission bomb
• Controlled fusion (for power plants) is still not achieved — main approaches: tokamak (magnetic confinement) and laser inertial confinement