Nuclei: Class 12 Physics · Chapter 13

13.1 Introduction

The sunlight warming your face left the Sun's core as energy set free when hydrogen nuclei joined to make helium. Some of the electricity in your home may come from Kudankulam or Tarapur, where uranium nuclei split. Both kinds of energy come from the nucleus (नाभिक), the tiny heart of the atom that the previous chapter uncovered with Rutherford's alpha-particle experiments.

Just how tiny is it? A typical atom is about ${10}^{-10}\mathrm{m}$ across, while its nucleus is about ${10}^{-14}\mathrm{m}$: ten thousand times smaller in radius. Volume goes as the cube of the radius, so the nucleus fills only about $({10}^{-4}{)}^3={10}^{-12}$ of the atom's volume. Picture an atom blown up to the size of a cricket stadium: the nucleus would be a grape sitting on the centre spot, and the electrons would roam the stands. Yet that grape carries almost all the mass. In a hydrogen atom the electron accounts for only 0.05 % of the mass; in heavier atoms the electrons' share is smaller still, so in every atom the nucleus holds over 99.9 % of the total mass.

This chapter asks four questions about that grape. What is it made of, and how do we write down its make-up? How big is it, and how dense? Why does it hold together at all, when its protons repel one another? And how can rearranging nuclei release energy on the scale of power stations and stars? Along the way we meet radioactivity, fission and fusion.

13.2 Atomic masses and composition of nucleus

A kilogram is a clumsy unit for atoms: one carbon atom has a mass of about $2\times {10}^{-26}\mathrm{k}\mathrm{g}$. Physicists and chemists instead use the atomic mass unit (symbol u), defined as one-twelfth of the mass of a single atom of carbon-12. We can find its value in kilograms from the mole. By definition of the mass unit, one mole of carbon-12 atoms has a mass of 12 g, and a mole holds Avogadro's number $N_A=6.022\times {10}^{23}$ of atoms. So one carbon-12 atom has a mass of $0.012\mathrm{k}\mathrm{g}\mathrm{/}{N}_A=1.9926\times {10}^{-26}\mathrm{k}\mathrm{g}$, and one-twelfth of that is the atomic mass unit.

Measured in u, the masses of atoms come out close to whole numbers, as if every atom were built from blocks of about the mass of a hydrogen atom. Some elements break the pattern badly: copper comes out at 63.55 u and magnesium at 24.31 u. The explanation came from the mass spectrometer, which sorts ions by their charge-to-mass ratio and so weighs individual atoms. It showed that most elements are mixtures of atoms with the same chemistry but different masses. These are called isotopes (समस्थानिक), from Greek words meaning same place, because they share one place in the periodic table. The atomic mass printed in a periodic table is an average of the isotope masses, weighted by how common each isotope is (its relative abundance).

Strip the single electron from an ordinary hydrogen atom and what is left is a bare proton. So the proton's mass follows from two measured numbers, the mass of the hydrogen atom (1.007825 u) and the mass of the electron, $m_e=0.000549\mathrm{u}$:

Ordinary hydrogen accounts for 99.985 % of the hydrogen in nature. Almost all the rest is deuterium (atomic mass 2.014102 u), found in every sample of water. A third isotope, tritium (3.016049 u), is unstable, with half of any sample decaying in about 12 years, so the tritium used in laboratories is manufactured.

How many protons does a nucleus hold? Count charges. A neutral atom of atomic number $Z$ has $Z$ electrons around it, carrying $-Ze$ in all, so its nucleus must carry $+Ze$. Each proton carries $+e$, and so the nucleus contains exactly $Z$ protons. Could some electrons be hiding inside the nucleus, cancelling the charge of extra protons? Physicists once suspected so, but quantum mechanics forbids it: an electron squeezed into a region a few femtometres across would have far more energy than any nucleus could hold on to.

Discovery of neutron

The hydrogen isotopes posed a puzzle. All three nuclei have charge $+e$, so each has just one proton, yet their masses stand roughly in the ratio 1 : 2 : 3. Deuterium and tritium nuclei must therefore carry extra neutral matter, about one and two proton masses of it. In 1932 James Chadwick identified what that neutral matter is (Fig. 13.2).

Earlier experimenters had seen that beryllium bombarded with alpha particles gives off a very penetrating, uncharged radiation, and that this radiation can kick protons out of paraffin wax at high speed. They took it to be gamma rays. Chadwick showed that this cannot be right. For a gamma-ray photon to give a proton that much kinetic energy, it would need several times more energy than the alpha-beryllium collision could supply. A neutral particle about as heavy as the proton fits naturally: a moving ball transfers its energy best to a target ball of the same mass, as in a head-on collision of billiard balls. Applying conservation of energy and momentum to protons and to nitrogen nuclei knocked on by the rays, Chadwick found the new particle's mass to be very close to the proton's. He called it the neutron, and received the 1935 Nobel Prize in Physics for the discovery. Its mass is now known very precisely:

The neutron is slightly heavier than the proton. A free neutron does not last: it turns into a proton, an electron and an antineutrino with a mean life of about 880 s, roughly a quarter of an hour. Bound inside a stable nucleus, the same neutron can last for ever.

We can now describe any nucleus by counting its particles. Three whole numbers do the job: $Z$ counts the protons (it is the atomic number of the element), $N$ counts the neutrons (the neutron number), and the mass number $A$ counts both together:

A nuclear species, or nuclide, is written ${\phantom{X}}_{\phantom{\mathrm{Z}}}^{\phantom{\mathrm{A}}}{\phantom{X}}_{\phantom{2}\mathrm{Z}}^{\phantom{2}\mathrm{A}}\mathrm{X}$, where X is the chemical symbol. For example, ${\phantom{X}}_{\phantom{82}}^{\phantom{208}}{\phantom{X}}_{\phantom{2}82}^{\phantom{2}208}\mathrm{P}\mathrm{b}$ is the most common nucleus of lead: 208 nucleons, of which 82 are protons and $208-82=126$ are neutrons. Since the symbol already fixes $Z$, people often drop the subscript and say lead-208 or write Pb-208.

Isotopes now have a simple meaning: nuclei with the same $Z$ but different $N$. Carbon-12, carbon-13 and carbon-14 all have 6 protons, with 6, 7 and 8 neutrons. Their atoms have identical electron arrangements and so the same chemistry, which is why chemists could not tell them apart. Tin holds the record, with ten stable isotopes. Two more families are useful. Isobars have the same mass number $A$: carbon-14, nitrogen-14 and oxygen-14 are isobars with $A=14$. Isotones have the same neutron number $N$: carbon-13, nitrogen-14 and oxygen-15 each have 7 neutrons. A chart with $N$ along one axis and $Z$ along the other shows all three families at a glance (Fig. 13.3).

13.3 Size of the nucleus

How do you measure something ${10}^{-14}\mathrm{m}$ across? You cannot look at it, so you throw things at it and watch how they bounce. In the Geiger–Marsden experiment of the previous chapter, alpha particles fired at gold foil were turned back by the electric repulsion of the gold nucleus. Rutherford's formula, which assumes nothing but Coulomb's law between two point charges, matched the results. That success already tells us something about size: the alphas never came close enough to feel anything other than electric repulsion.

In a head-on approach, the alpha particle slows down as its kinetic energy $K$ turns into electric potential energy. It stops for an instant at the distance of closest approach $d$, where all of $K$ has become potential energy, and then turns back. With charge $2e$ on the alpha particle and $Ze$ on the nucleus,

Since the nuclear charge cannot be spread wider than the region the alpha never entered, $d$ is an upper limit on the radius (Fig. 13.5). A handy number for such sums is $e^2\mathrm{/}4\pi {\epsilon }_0=1.44\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{f}\mathrm{m}$, where $1\mathrm{f}\mathrm{m}={10}^{-15}\mathrm{m}$ (one femtometre, also called a fermi).

With faster projectiles the story changes. Once the projectile reaches the edge of the nucleus, the short-range nuclear force joins in and the scattering stops following Rutherford's formula; the energy at which this deviation begins gives the nuclear size. Even better probes are fast electrons. An electron does not feel the nuclear force at all, so its scattering maps out how the nuclear charge (the protons) is spread. Alpha particles and neutrons, which do feel the nuclear force, map the nuclear matter instead, so the two methods give slightly different radii. Experiments of both kinds, on many elements, agree on a remarkably simple rule:

The cube root is the clue. The volume of a sphere goes as $R^3$, so the volume of a nucleus goes as $R_0^{3}A$: it is simply proportional to the number of nucleons. Doubling the nucleons doubles the volume, but stretches the radius only by $2^{1\mathrm{/}3}=1.26$ (Fig. 13.6). Nucleons in a nucleus are packed like marbles in a jar or molecules in a drop of water: adding more makes a bigger drop of the same density. This liquid-drop picture of the nucleus will help us understand fission later.

13.4 Mass-energy and nuclear binding energy

13.4.1 Mass – Energy

Before 1905, physics had two separate bookkeeping rules: the total mass in a reaction stays the same, and so does the total energy. Einstein's special theory of relativity merged them. Mass, he showed, is itself a form of energy, and it can be turned into kinetic energy, light or heat, and back again. The exchange rate is fixed by the speed of light $c$ in vacuum:

Because $c^2$ is about $9\times {10}^{16}{\mathrm{m}}^2{\mathrm{s}}^{-2}$, a little mass is worth an enormous amount of energy.

Nobody can turn dust wholly into energy, but nuclear reactions and the creation and annihilation of particles test Eq. 13.6 directly, and it has passed every test. The modern statement is a single law: in any isolated process the total energy is conserved, provided we count the energy $m{c}^2$ of every mass. There are no longer separate laws for mass and energy, only one law of conservation of mass–energy. Nuclear physics gives the most striking evidence for it, and the rest of this chapter runs on it.

13.4.2 Nuclear binding energy

A nucleus is made of protons and neutrons, so you might expect its mass to equal the masses of its parts added up. Weigh them and it does not. Take carbon-12, whose atom has a mass of exactly 12.000000 u by definition. It has 6 protons, 6 neutrons and 6 electrons. A neat trick handles the electrons: count the protons and electrons together as 6 hydrogen atoms, so that the electrons appear on both sides and cancel. The parts then weigh

The assembled atom is lighter by $12.098940-12.000000=0.098940\mathrm{u}$, almost a tenth of a nucleon's mass (Fig. 13.7). Every nucleus shows such a shortfall, called its mass defect (द्रव्यमान क्षति). For a nucleus of mass $M$ with $Z$ protons and $A-Z$ neutrons,

With atomic masses from tables, replace $m_p$ by $m_{\mathrm{H}}$ and $M$ by the atomic mass, as we did for carbon; the electron masses cancel. (The tiny binding energies of the electrons themselves, a few keV at most, are ignored.)

Where has the mass gone? Mass–energy equivalence answers this. When 6 free protons and 6 free neutrons come together to form a carbon nucleus, they give out energy, and that energy carries off mass $\mathrm{\Delta }M$. Run the process backwards: to pull the nucleus apart into separate nucleons you must supply the same energy, and the separated parts are then heavier by exactly $\mathrm{\Delta }M$. This energy is the binding energy (बंधन ऊर्जा) of the nucleus:

Binding energy is a bookkeeping quantity: experiments rarely smash a nucleus into single nucleons, yet comparing binding energies tells us which nuclei are held more firmly and which reactions can release energy. To use Eq. 13.8 with masses in u, we need to know what one atomic mass unit is worth in energy.

For carbon-12 the same conversion gives $E_b=0.098940\times 931.5\mathrm{M}\mathrm{e}\mathrm{V}=92.2\mathrm{M}\mathrm{e}\mathrm{V}$. Big nuclei naturally have big binding energies simply because they have more nucleons, so to compare nuclei fairly we divide by $A$. The binding energy per nucleon,

is each nucleon's share of the binding: the total cost of dismantling the nucleus, divided equally among its $A$ members. Carbon-12 has $92.2\mathrm{/}12=7.68\mathrm{M}\mathrm{e}\mathrm{V}$ per nucleon. Work this out for every stable nucleus, plot it against the mass number, and a famous curve appears (Fig. 13.8).

Read the curve from left to right. For the lightest nuclei $E_{bn}$ is small (1.1 MeV for the deuteron) and rises steeply, with a few nuclei sitting noticeably above their neighbours: helium-4 (7.07 MeV), carbon-12 and oxygen-16. These favoured nuclei are evidence that nucleons fill shells inside the nucleus, much as electrons fill shells in an atom. Beyond about $A=30$ the curve flattens out between 8 and 8.8 MeV per nucleon all the way to $A\approx 170$, with its highest point around iron-56 and nickel-62 at about 8.8 MeV. For the heaviest nuclei it sags slowly, to 7.57 MeV for uranium-238.

Four conclusions follow, and they shape the rest of the chapter.

1. The binding force is strongly attractive. Holding each nucleon with several MeV, against the electric repulsion of the protons, needs a force much stronger than any we have met so far.
2. The force has a short range. If every nucleon attracted every other one, each new nucleon would add to the binding of all the rest, and $E_{bn}$ would keep growing with $A$. Instead it stays flat, which means a nucleon interacts only with the few neighbours within reach.
3. Splitting a heavy nucleus releases energy. Its nucleons, bound by about 7.6 MeV each, end up in middle-sized nuclei where they are bound by about 8.4 to 8.7 MeV. Every nucleon becomes more tightly bound, and the difference comes out as energy: this is fission (Section 13.7.1).
4. Joining light nuclei releases energy. Two deuterons (1.1 MeV per nucleon) that combine into helium-4 (7.1 MeV per nucleon) end up far more tightly bound. This is fusion, the energy source of the Sun (Section 13.7.2).

The second conclusion deserves a closer look (Fig. 13.9). Suppose the pull reaches only about 2 fm, roughly as far as the nearest neighbours. Then a nucleon buried inside a nucleus is surrounded by the same small crowd, about a dozen nucleons in three dimensions, whether the nucleus has 40 nucleons or 200. Nucleons added on the far side are out of reach and change nothing for it. In a large nucleus most nucleons are buried in this way, so $E_{bn}$ settles to a constant. Only the nucleons at the surface, with fewer neighbours, are bound less; in a small nucleus nearly all nucleons are at the surface, which is why $E_{bn}$ is lower for light nuclei. This property of a short-range force, that each nucleon is held only by a limited number of neighbours however large the nucleus, is called saturation.

13.5 Nuclear force

Electrons are held in atoms by the Coulomb force; the outer electrons, which decide chemistry, are bound by only a few eV. Nucleons are held with about 8 MeV each, roughly a million times more, so a different force must be at work. Its job is hard. Two protons 2 fm apart repel each other with a force of $F=e^2\mathrm{/}(4\pi {\epsilon }_0{r}^2)\approx 58\mathrm{N}$, the weight of a 6 kg mass, acting on a particle of mass ${10}^{-27}\mathrm{k}\mathrm{g}$. Their electric potential energy at that distance is about 0.72 MeV. Gravity is no help: between the same two protons it is about ${10}^{36}$ times weaker than the electric push. The force that wins is called the nuclear force (नाभिकीय बल), or the strong force. Experiments carried out mostly between 1930 and 1950 established its main features.

1. It is much stronger than the Coulomb force at nuclear distances, strong enough to bind protons that repel one another into a volume a few femtometres across. Gravity, in turn, is far weaker than both.
2. It has a short range. Between two nucleons the force fades away once they are more than a few femtometres apart. This is the origin of saturation and of the flat stretch of the binding-energy curve.
3. It is attractive at moderate distances but strongly repulsive at very short ones. The potential energy of two nucleons is lowest at a separation $r_0$ of about 0.8 fm. Farther apart than $r_0$, they attract; pushed closer than $r_0$, they repel very strongly. This hard core stops nuclei from collapsing.
4. It does not depend on charge. The nuclear pull between two protons, two neutrons, or a proton and a neutron is about the same.

The force on a nucleon is the negative slope of the potential-energy curve, $F=-dU\mathrm{/}dr$. Where $U$ rises as $r$ grows (beyond $r_0$), $F$ is negative, meaning it points towards smaller $r$: attraction. Where $U$ rises as $r$ shrinks (inside $r_0$), the force points outwards: repulsion. Unlike Coulomb's law or Newton's law of gravitation, the nuclear force has no simple formula, and the curve in Fig. 13.10 is only a sketch of its shape.

13.6 Radioactivity

In February 1896 Henri Becquerel was testing whether uranium salts, after soaking up sunlight, give off penetrating rays like the X-rays discovered a few months earlier. A run of cloudy days in Paris stopped his experiment, so he left the salts on a photographic plate wrapped in black paper inside a drawer. When he developed the plate anyway, it was strongly fogged. The uranium was emitting radiation all by itself, with no sunlight needed. Marie and Pierre Curie soon found two far more active elements, polonium and radium, and Marie Curie named the effect radioactivity (रेडियोऐक्टिवता).

Later experiments showed that radioactivity comes from the nucleus. An unstable nucleus changes into another nucleus, or into a lower-energy state of itself, and throws out radiation in the process. This is radioactive decay. Three kinds occur in nature, named with the first three letters of the Greek alphabet before anyone knew what they were:

• Alpha ($\alpha$) decay: the nucleus emits a helium-4 nucleus, ${\phantom{X}}_{\phantom{2}}^{\phantom{4}}{\phantom{X}}_{\phantom{2}2}^{\phantom{2}4}\mathrm{H}\mathrm{e}$, two protons and two neutrons bound together.
• Beta ($\beta$) decay: the nucleus emits an electron (${\beta }^{-}$) or a positron (${\beta }^{+}$), a particle with the electron's mass but charge $+e$. A neutrino or antineutrino leaves too.
• Gamma ($\gamma$) decay: an excited nucleus drops to a lower energy state and emits a photon of high energy, typically hundreds of keV or more.

In every decay, charge and the total number of nucleons are conserved: add up the top numbers and the bottom numbers on each side of the examples in Table 13.1 and they match. Inside the nucleus, ${\beta }^{-}$ decay turns a neutron into a proton, and ${\beta }^{+}$ decay turns a proton into a neutron. On a nuclide chart each decay is a fixed step (Fig. 13.11).

Why are some nuclei unstable at all? Light stable nuclei have about as many neutrons as protons ($N\mathrm{/}Z\approx 1$). Heavier ones need extra neutrons: their many protons repel each other over the whole nucleus, while the nuclear attraction reaches only nearest neighbours, and extra neutrons add attraction without adding repulsion. By lead-208 the ratio is $126\mathrm{/}82\approx 1.5$. A nucleus with too many neutrons for its size tends to undergo ${\beta }^{-}$ decay, one with too many protons ${\beta }^{+}$ decay, and very heavy nuclei often shed an alpha particle. Stability is the exception: of the more than 3000 nuclides known, fewer than one in ten is stable. The rest are made in laboratories by bombarding stable nuclei with protons, neutrons, deuterons or alpha particles, or are seen in the debris of stars.

13.7 Nuclear energy

The binding-energy curve is also a map of where energy can be found. Its middle is a long plateau at about 8 MeV per nucleon, and both ends lie lower. A system that is more tightly bound has less mass, so whenever nucleons rearrange from loosely bound nuclei into more tightly bound ones, mass disappears and energy comes out. There are two ways to climb towards the plateau: split a very heavy nucleus into two middle-sized ones (fission), or join two very light nuclei into a heavier one (fusion).

The energy given out in a nuclear process is its Q-value. By conservation of mass–energy it can be found either from the kinetic energies or from the masses:

A positive $Q$ means energy is released (the process is exothermic); a negative $Q$ means energy must be supplied (endothermic). Since protons and neutrons are only regrouped, $Q$ also equals the total binding energy after minus the total binding energy before.

Compare this with chemistry. Coal, petrol and cooking gas release energy by rearranging electrons between atoms, a few eV per atom: burning one carbon atom into carbon dioxide gives about 4.1 eV. A single uranium fission gives about 200 MeV, roughly fifty million times more. Per kilogram, the complete fission of uranium-235 would give about $8\times {10}^{13}\mathrm{J}$, while a kilogram of good coal gives about $3\times {10}^7\mathrm{J}$. Kilogram for kilogram, nuclear fuel is a few million times richer than chemical fuel. Explore the curve and both kinds of reaction in the lab below.

13.7.1 Fission

Natural radioactivity is slow and cannot be switched on or off. Far more control comes from nuclear reactions, in which we fire particles such as protons, deuterons, alpha particles or neutrons at nuclei. Neutrons make the best bullets: being uncharged, they feel no Coulomb repulsion and can reach a nucleus even when moving slowly. In 1938 Otto Hahn and Fritz Strassmann bombarded uranium with neutrons and, to their surprise, found barium among the products, an element about half as heavy. Lise Meitner and Otto Frisch explained it within weeks: the uranium nucleus had split in two. They called this nuclear fission (नाभिकीय विखंडन).

When uranium-235 absorbs a slow neutron, it becomes uranium-236 in a highly excited state, written ${\phantom{X}}_{\phantom{92}}^{\phantom{236}}{\phantom{X}}_{\phantom{2}92}^{\phantom{2}236}\mathrm{U}{\phantom{X}}^{\ast }$. This nucleus wobbles like a struck water drop and, in most cases, splits into two medium-sized fragments and a few neutrons (Fig. 13.13):

The split is not the same every time. Three of the many ways the excited nucleus can break up are

In each case the charges balance too: barium, krypton, xenon, strontium, caesium and rubidium have $Z=56,36,54,38,55$ and $37$, and each pair adds up to uranium's 92.