Nuclei

Composition of the nucleus

• More than $99.9\%$ of an atom's mass sits in its nucleus, whose radius is smaller than the atom's by a factor of about $10^{4}$; the nuclear volume is about $10^{-12}$ of the atomic volume. An atom is almost entirely empty space.
• The nucleus contains protons and neutrons only. Electrons are never inside the nucleus — quantum arguments rule it out; all $Z$ electrons are outside.
• $Z$ = number of protons = atomic number, $N$ = number of neutrons, $A = Z + N$ = mass number = number of nucleons. Nuclear charge $= +Ze$.
• A nuclide is written ${}^{A}_{Z}\text{X}$. Example: ${}^{197}_{79}\text{Au}$ — 197 nucleons, 79 protons, 118 neutrons.
• The atomic mass unit: $1\ \text{u} = \dfrac{1}{12}\times$ (mass of one ${}^{12}_{6}\text{C}$ atom) $= 1.660539\times10^{-27}$ kg.
• Proton: $m_p = 1.00727$ u, charge $+e$, stable. Neutron: $m_n = 1.00866$ u, uncharged, discovered by James Chadwick in 1932 (Nobel Prize 1935) by bombarding beryllium with $\alpha$-particles.
• A FREE neutron is unstable: it decays into a proton, an electron and an antineutrino with a mean life of about 1000 s. Inside a nucleus it is stable.
• Atomic masses are measured with a mass spectrometer.

Isotopes, isobars, isotones

• Isotopes have identical electronic structure, hence identical chemical behaviour, and occupy the same place in the periodic table — which is what the word means.
• Most elements are mixtures of isotopes; the quoted atomic mass is the weighted average over their relative abundances. Chlorine: $34.98$ u ($75.4\%$) and $36.98$ u ($24.6\%$) average to $35.47$ u.
• Hydrogen's three isotopes: protium $1.0078$ u ($99.985\%$), deuterium $2.0141$ u, tritium $3.0160$ u. Tritium nuclei are unstable and are made artificially.

Size and density of the nucleus

• Nuclear radius: $R = R_0 A^{1/3}$ with $R_0 = 1.2$ fm. Measured by scattering fast electrons (and, less directly, $\alpha$-particles) off nuclei of many elements.
• Since $R \propto A^{1/3}$, $V \propto R^3 \propto A$, so the nuclear density $\rho = \dfrac{3m}{4\pi R_0^3}$ contains no $A$ and is the SAME for every nucleus, $\approx 2.3\times10^{17}$ kg m$^{-3}$.
• That is about $10^{14}$ times the density of water ($10^{3}$ kg m$^{-3}$) — because ordinary matter is mostly empty space while a nucleus is not. Nuclei behave like drops of a liquid of constant density; the matter in neutron stars has a comparable density.
• Electron scattering senses the CHARGE distribution of a nucleus, $\alpha$-scattering senses the nuclear MATTER distribution, so the two radii come out slightly different.
• Rutherford, Geiger and Marsden: a $5.5$ MeV $\alpha$-particle approaches a gold nucleus to about $4.0\times10^{-14}$ m, so the nucleus must be smaller than that. At higher $\alpha$ energies the scattering starts to depart from the pure Coulomb prediction, and the energy at which it does gives the nuclear size.

Mass–energy equivalence

• $E = mc^2$: mass is a form of energy. Conservation of mass and conservation of energy are not separate laws — only their combination holds.
• Energy equivalent of 1 u: $1\ \text{u} = 931.5\ \text{MeV}/c^2$. Energy equivalent of 1 g: $9\times10^{13}$ J.
• Mass–energy interconversion happens in chemical reactions too, but the mass defects there are about a million times smaller, which is why it goes unnoticed.

Mass defect and binding energy

• The measured mass $M$ of a nucleus is always LESS than the total mass of its free constituents. The shortfall is the mass defect

$$\Delta M = \left[Z m_p + (A-Z)m_n\right] - M$$

• Physical reason (a 1-mark favourite): in forming the nucleus the nucleons are brought together against their mutual Coulomb repulsion and bound by the strong nuclear force; the binding energy released comes out of their rest mass.
• Binding energy $E_b = \Delta M\,c^2$: the energy released on forming the nucleus, equal to the energy needed to break it back into free nucleons. In MeV, $E_b = \Delta M(\text{u})\times931.5$.
• Worked standard: for ${}^{16}_{8}\text{O}$, $\Delta M = 0.13691$ u, so $E_b = 127.5$ MeV.
• Binding energy per nucleon $E_{bn} = E_b/A$ is the better measure of stability — the average energy needed to pull one nucleon out.
• If the data are ATOMIC masses, use $m_H$ in place of $m_p$; the electron masses cancel automatically.

The binding-energy-per-nucleon curve, in words

• Plot $E_{bn}$ (MeV per nucleon, vertical) against mass number $A$ (horizontal). Shape: a steep rise from nearly zero at $A = 1$, sharp local peaks at ${}^{4}\text{He}$, ${}^{12}\text{C}$, ${}^{16}\text{O}$, a broad maximum of about $8.75$ MeV near $A = 56$ (iron), then a long slow fall to $7.6$ MeV at $A = 238$ (uranium) — a shallow, asymmetric hump.
• Feature (i): $E_{bn}$ is practically constant, i.e. practically independent of $A$, over the middle range $30 < A < 170$, at about $8.0$ MeV per nucleon.
• Feature (ii): $E_{bn}$ is lower for both light nuclei ($A < 30$) and heavy nuclei ($A > 170$).
• What the size of $E_{bn}$ implies: the nuclear force is attractive and strong enough to bind nucleons by several MeV each.
• What the flatness implies: the nuclear force is short-ranged and SATURATES — a nucleon inside a large nucleus is attracted only by the few neighbours within range, so adding more nucleons does not increase the binding per nucleon.
• What the fall at large $A$ implies: a nucleus with $A = 240$ is less tightly bound than two with $A = 120$, so splitting it releases energy — FISSION.
• What the rise at small $A$ implies: two very light nuclei ($A \le 10$) joining make a more tightly bound product, so combining them releases energy — FUSION.
• The peaks at ${}^{4}\text{He}$, ${}^{16}\text{O}$ and similar nuclides on an otherwise smooth curve are taken as evidence of an atom-like shell structure inside nuclei.

Nuclear force

• The force binding nucleons is not Coulombic and not gravitational; it is a distinct strong attractive interaction. Its scale is set by $E_{bn} \approx 8$ MeV, far above atomic binding energies of a few eV.
• Property 1 — strength: much stronger than the Coulomb force between charges and enormously stronger than gravity. It must be, to overwhelm the proton–proton repulsion inside the tiny nuclear volume.
• Property 2 — short range: it falls rapidly to zero once the separation exceeds a few femtometres. This short range is the cause of saturation, and saturation is the cause of the constancy of $E_{bn}$ for $30 < A < 170$.
• Property 3 — charge independence: the nuclear force between n–n, p–n and p–p pairs is approximately the same. It does not depend on electric charge.
• Property 4 — a repulsive core: on the potential-energy curve the minimum ($\approx -100$ MeV) lies at $r_0 \approx 0.8$ fm. For $r > r_0$ the force is attractive; for $r < r_0$ it is strongly repulsive. So "nuclear forces are always attractive" is FALSE.
• There is no simple mathematical formula for the nuclear force, unlike Coulomb's law or Newton's law of gravitation.

Radioactivity

• Radioactivity was discovered by A. H. Becquerel in 1896, by accident: uranium–potassium sulphate wrapped in black paper blackened a photographic plate through a silver sheet.
• Radioactivity is a NUCLEAR phenomenon in which an unstable nucleus undergoes decay; it is an indication of nuclear instability.
• Three types of decay occur in nature: $\alpha$-decay, in which a helium nucleus ${}^{4}_{2}\text{He}$ is emitted; $\beta$-decay, in which electrons or positrons are emitted; and $\gamma$-decay, in which high-energy photons (hundreds of keV or more, wavelengths shorter than X-rays) are emitted.
• Stability needs a neutron-to-proton ratio near $1:1$ for light nuclei, rising to about $3:2$ for heavy nuclei — extra neutrons are needed to offset the growing proton–proton repulsion. Nuclei far from this ratio are unstable. Only about $10\%$ of known isotopes are stable.
• An electron and a positron are a particle–antiparticle pair: equal masses, opposite charges. They annihilate on meeting, giving $\gamma$-ray photons.

Nuclear energy: general principle

• The greater the binding energy of a bound system, the less its total mass. So if nuclei with smaller total binding energy transform into nuclei with greater total binding energy, energy is released.
• Two ways to move towards the peak of the $E_{bn}$ curve: a heavy nucleus splits into intermediate-mass fragments (fission), or light nuclei join into a heavier one (fusion).
• Nuclear energies are of order MeV against a few eV for chemical reactions, so for the same quantity of matter a nuclear source yields about a million times more energy: $10^{14}$ J per kg of uranium fissioned, against $10^{7}$ J per kg of coal burnt.
• $Q$-value: $Q = (\sum m_{\text{initial}} - \sum m_{\text{final}})c^2$ = (final KE) $-$ (initial KE). Positive $Q$ means exothermic and possible; negative $Q$ means endothermic and not spontaneously possible.
• Nuclear reactions are balanced in the sense that mass number and charge number are separately conserved on the two sides — not in the chemical sense that the number of atoms of each element is conserved, because elements are transmuted.

Nuclear fission

• Fission is the breaking of a heavy nucleus, when bombarded by a neutron, into two intermediate-mass fragments with the release of energy and two to four neutrons.

$${}^{1}_{0}\text{n} + {}^{235}_{92}\text{U} \to {}^{236}_{92}\text{U} \to {}^{144}_{56}\text{Ba} + {}^{89}_{36}\text{Kr} + 3\,{}^{1}_{0}\text{n}$$

• The fragments are themselves radioactive and reach stable end products by a succession of $\beta$ emissions.
• Energy released $\approx 200$ MeV per fissioning nucleus. NCERT's estimate: $A = 240$ at $7.6$ MeV/nucleon into two $A = 120$ at $8.5$ MeV/nucleon gives a gain of $0.9$ MeV per nucleon, so $240\times0.9 = 216$ MeV.
• The energy first appears as kinetic energy of the fragments and neutrons, and is then transferred to the surroundings as heat.
• Controlled fission in a nuclear reactor produces electricity; uncontrolled fission is the atom bomb.
• Why very heavy nuclei fission spontaneously: they hold so many protons that the total Coulomb repulsion is huge, while the short-range nuclear attraction saturates and cannot keep pace, so splitting into two more tightly bound fragments lowers the energy.

Nuclear fusion and energy generation in stars

• Fusion is the combining of two light nuclei into a single heavier nucleus with the release of energy, because the product is more tightly bound.
• Examples: ${}^{1}_{1}\text{H} + {}^{1}_{1}\text{H} \to {}^{2}_{1}\text{H} + e^{+} + \nu + 0.42$ MeV; ${}^{2}_{1}\text{H} + {}^{2}_{1}\text{H} \to {}^{3}_{2}\text{He} + n + 3.27$ MeV; ${}^{2}_{1}\text{H} + {}^{2}_{1}\text{H} \to {}^{3}_{1}\text{H} + {}^{1}_{1}\text{H} + 4.03$ MeV.
• Fusion needs the nuclei to come within range of the nuclear force, but both are positively charged, so they must first overcome the Coulomb barrier — about $400$ keV for two protons, higher for larger charges.
• Raising the temperature so that the particles have enough kinetic energy to cross that barrier gives THERMONUCLEAR fusion; $\dfrac32 kT = 400$ keV gives $T \sim 3\times10^{9}$ K.
• The Sun's core is at $1.5\times10^{7}$ K, well below that, so solar fusion is carried by the small fraction of protons with energies far above the average.
• The proton–proton cycle, in four steps, burns hydrogen into helium; the net effect of $2\times$(i)$+ 2\times$(ii)$+ 2\times$(iii)$+$(iv) is $4\,{}^{1}_{1}\text{H} + 2e^{-} \to {}^{4}_{2}\text{He} + 2\nu + 6\gamma + 26.7$ MeV.
• The Sun is about $5\times10^{9}$ years old with hydrogen for roughly another 5 billion years; then it will cool, collapse, heat its core, expand its envelope and become a red giant.
• At about $10^{8}$ K helium fuses to carbon, and successively heavier elements can be built this way — but only up to the peak of the $E_{bn}$ curve. Elements much heavier than iron cannot be produced by stellar fusion.
• Controlled thermonuclear fusion on Earth: heat the fuel to about $10^{8}$ K, at which it is a plasma of positive ions and electrons. The central problem is confinement, since no material container survives such a temperature.

Fission compared with fusion

Numbers worth memorising

Traps that cost marks

• Writing the radius law as $R = R_0 A^{3}$ or $R = R_0 A$. It is $R = R_0 A^{1/3}$ — the cube ROOT of $A$.
• Being asked for a ratio of nuclear densities and computing something from $A$. Nuclear density is independent of $A$: every such ratio is $1:1$, whatever the mass numbers.
• Confusing mass defect with binding energy. $\Delta M$ is a mass (in u or kg); $E_b = \Delta M c^2$ is an energy (in J or MeV).
• Multiplying by $931.5$ AND by $c^2$. The factor $931.5$ MeV per u already contains $c^2$.
• Mixing nuclear and atomic masses in one mass-defect calculation. If the table gives atomic masses, use $m_H = 1.007825$ u (not $m_p$) and the atomic mass of the nuclide; the electrons then cancel.
• Saying "nuclear forces are always attractive". Below $r_0 \approx 0.8$ fm they are strongly repulsive — that repulsive core is what stops the nucleus collapsing.
• Saying the nuclear force is weaker than the Coulomb force. It is much stronger; that is precisely why a nucleus full of protons holds together.
• Attributing the constancy of $E_{bn}$ to charge independence. It is due to SATURATION, which follows from the short range of the nuclear force.
• Putting the maximum of the $E_{bn}$ curve at large $A$. The maximum ($\approx 8.75$ MeV) is at $A = 56$; heavy nuclei sit lower, at $7.6$ MeV for $A = 238$.
• In a "how long will the lamp glow" numerical, forgetting that ONE fusion event consumes TWO deuterium nuclei — the energy per deuterium nucleus is $3.27/2$ MeV.
• Using molar mass 1 instead of 2 g mol$^{-1}$ for deuterium when counting nuclei.
• Getting the sign of $Q$ from binding energies backwards. $Q =$ B.E. of products $-$ B.E. of reactants; from masses it is $\sum m_{\text{initial}} - \sum m_{\text{final}}$.
• Concluding a reaction "releases energy" when $Q$ comes out negative. A negative $Q$ means energy must be supplied and the process is not energetically possible.
• Saying nucleon number is not conserved when energy is released. Proton number and neutron number are both conserved; what changes is the total BINDING energy, and that difference appears as the released energy.
• Saying the nucleus contains electrons because it emits them in $\beta$-decay. All the electrons of an atom are outside the nucleus.
• Calling a free neutron stable. It decays with a mean life of about 1000 s; only inside a nucleus is it stable.
• Balancing a nuclear reaction like a chemical one. Balance mass number and charge number separately; elements may be transmuted.
• Writing "fermi" as anything other than $10^{-15}$ m, and forgetting that 1 fermi = 1 femtometre.