Structure of Atoms and Nuclei

15.1 Introduction

• Leucippus and Democritus first proposed indivisible atoms; Dalton's theory said matter is made of indestructible particles, atoms of a given element are identical, and atoms combine to form new substances.
• J. J. Thomson's discovery of the electron destroyed "indestructible". Thomson's model was improved by Rutherford, and Rutherford's by Bohr.
• Two numbers that frame the chapter: the nucleus is about $10^5$ times smaller in radius than the atom, and it carries all the positive charge and $99.9\%$ of the mass.

15.2 Thomson's atomic model

• Evidence: in an evacuated discharge tube with a voltage across two electrodes, the cathode emits rays that glow on striking the glass behind the anode; the rays are negatively charged particles — electrons.
• Model (1903): the atom is a sphere of uniform positive charge in which electrons are embedded — the plum-pudding model.
• Successes: the atom is electrically neutral (total positive charge = total negative charge); only electrons can be emitted, since the positive charge is spread through the whole solid sphere; it explains the formation of ions and ionic compounds.
• Failure: the alpha-scattering experiment showed the positive charge is not spread out at all.

15.3 Geiger-Marsden experiment

• Suggested by Rutherford, performed by Geiger and Marsden between 1908 and 1913. Alpha particles (helium nuclei, charge $+2e$) from a radioactive source were collimated into a narrow beam and fired at a thin gold foil; scattered particles produced scintillations on a surrounding screen viewed through a movable microscope.
• Observations: most alpha particles went straight through; a few were scattered through various angles; only about $0.14\%$ were scattered through more than $0.1^\circ$; about one in $8000$ was deflected through more than $90^\circ$; a very few came back through nearly $180^\circ$.
• Numerical use: for a head-on approach, the initial kinetic energy equals $\dfrac{1}{4\pi\varepsilon_0}\dfrac{2Ze^2}{r_1+r_2}$ at the point where the alpha particle stops.

15.4 Rutherford's atomic model

• Thomson's model cannot explain the results: a diffuse positive charge has a small volume density and could only produce small deflections.
• Rutherford's reasoning: a particle turned back must have met something massive and highly positively charged; since only a very few turn back, that something must be very small. Experiment gave its size as about $10\ \text{fm}$, roughly $10^{-5}$ times the size of the atom, i.e. about $10^{-15}$ of its volume. He called it the nucleus.
• The model: all the positive charge and $99.9\%$ of the mass sit in a central nucleus; electrons revolve around it in circular orbits like planets round the Sun; the space between is mostly empty, which is why most alpha particles pass undeflected.
• It also explains why atoms emit electrons but not positive particles — the heavy nucleus is unaffected by ordinary forces on the atom.

Difficulties with Rutherford's model

• An electron in a circular orbit is accelerated (its direction changes), and by Maxwell's equations an accelerated charge radiates continuously.
• Radiating, it loses energy, its orbital radius shrinks continuously, and it should spiral into the nucleus — the atom would collapse.
• As it spirals in, its speed and hence the frequency of the radiation would change continuously, giving a continuous spectrum.
• None of this is observed: atoms are stable, they do not radiate on their own, and when excited they emit only certain fixed frequencies. It also could not explain atomic spectra.

15.5 Atomic spectra

• A heated metallic object gives a continuous spectrum through a prism. Hydrogen gas heated in a glass tube gives a line spectrum — only a few discrete wavelengths, with nothing in between.
• Four visible hydrogen lines: $410$, $434$, $486$ and $656\ \text{nm}$. These lines are called emission lines.
• Lines fall into series, named from shortest to longest wavelength: Lyman, Balmer, Paschen, Brackett, Pfund.
• Within a series the separation of successive lines decreases as the wavelength gets shorter, until they crowd up to the series limit $\dfrac{n^2}{R}$ reached as $m\to\infty$.
• Empirical relation: $\dfrac{1}{\lambda} = R\left(\dfrac{1}{n^2}-\dfrac{1}{m^2}\right)$, with $n$ fixed for a series and $m > n$.
• Every element gives its own unique set of wavelengths, so an element can be identified from its spectrum.

15.6 Bohr's atomic model

• Bohr kept Rutherford's picture and added quantum ideas to make the orbits stable and non-radiating.
• Postulate 1: the electron revolves round the nucleus in a fixed circular orbit with constant speed, the electrostatic attraction providing the centripetal force — $\dfrac{m_ev_n^2}{r_n} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Ze^2}{r_n^2}$.
• Postulate 2: only orbits whose angular momentum is an integral multiple of $\dfrac{h}{2\pi}$ are allowed — $m_e v_n r_n = \dfrac{nh}{2\pi}$. Electrons in these stable orbits do not radiate.
• Postulate 3: an electron may jump to an orbit of lower energy, emitting one photon of energy equal to the difference — $E_m - E_n = h\nu$.
• Standard results: $r_n = \left(\dfrac{\varepsilon_0h^2}{\pi m_eZe^2}\right)n^2 = a_0n^2$ with $a_0 = 0.053\ \text{nm}$; $v_n = \dfrac{Ze^2}{2\varepsilon_0nh}$; $E_n = -\dfrac{m_eZ^2e^4}{8\varepsilon_0^2h^2n^2} = -\dfrac{13.6Z^2}{n^2}\ \text{eV}$.
• Energy-level facts for hydrogen: $E_1 = -13.6$, $E_2 = -3.4$, $E_3 = -1.51$, $E_4 = -0.85$, $E_5 = -0.544\ \text{eV}$; levels crowd together and reach $0$ as $n\to\infty$.
• Three definitions the board asks together — excitation energy: energy needed to lift the electron from the ground state to a given excited state (minimum $10.2\ \text{eV}$ for hydrogen). Ionisation energy: minimum energy needed to set the electron free from the state it is in ($13.6\ \text{eV}$ from the ground state). Binding energy: the energy with which the electron is bound, equal in magnitude to its total energy, $B.E = -T.E$.
• Rydberg's formula follows from postulate 3: $\bar\nu = \dfrac{1}{\lambda} = R_HZ^2\left(\dfrac{1}{n^2}-\dfrac{1}{m^2}\right)$, with $R_H = \dfrac{m_ee^4}{8\varepsilon_0^2h^3c} = 1.097\times10^7\ \text{m}^{-1}$.
• Useful standard result: the frequency of the first Lyman line, $R_Hc\left(1-\dfrac{1}{4}\right) = \dfrac{3R_Hc}{4}$, equals the difference of the limiting frequencies of the Lyman ($R_Hc$) and Balmer ($\dfrac{R_Hc}{4}$) series.

Limitations of Bohr's model

• It cannot explain the spectra of atoms other than hydrogen, nor the fine multiple components of some hydrogen lines.
• It gives no account of why emission lines differ in intensity.
• It is theoretically unsatisfactory: the quantisation condition for stable orbits is simply assumed, with no basis.

De Broglie's explanation of the second postulate

• Treat the orbiting electron as a standing wave rather than a particle. A standing wave fits only if the orbit length is a whole number of wavelengths: $2\pi r_n = n\lambda_n$.
• With $\lambda_n = \dfrac{h}{p_n}$ this gives $p_n = \dfrac{nh}{2\pi r_n}$, hence $L_n = p_nr_n = \dfrac{nh}{2\pi}$ — exactly Bohr's second postulate.
• Diagram to describe: a circular orbit with a continuous wavy line drawn along it closing smoothly on itself; for $n=4$, four complete wavelengths fit round the circle.

15.7 Atomic nucleus

• The nucleus is made of protons and neutrons, together called nucleons. A proton carries $+e$; a neutron is neutral and slightly heavier. A proton is about $1836$ times as massive as an electron.
• $Z$ = number of protons = number of electrons in the neutral atom = atomic number; $N$ = number of neutrons; $A = Z+N$ = mass number. Notation ${}^{A}_{Z}\text{X}$, e.g. ${}^{1}_{1}\text{H}$, ${}^{12}_{6}\text{C}$, ${}^{16}_{8}\text{O}$.
• Chemical properties are decided by $Z$ alone.

Units of mass

• Kilogram: $m_e = 9.109383\times10^{-31}$, $m_p = 1.672623\times10^{-27}$, $m_n = 1.674927\times10^{-27}\ \text{kg}$.
• Unified atomic mass unit: $1\ \text{u}$ is one twelfth of the mass of a neutral carbon-12 atom in its lowest electronic state, $= 1.6605402\times10^{-27}\ \text{kg}$. In this unit $m_e = 0.00055$, $m_p = 1.00728$, $m_n = 1.00866\ \text{u}$, and the hydrogen atom is $1.007825\ \text{u}$.
• Energy equivalent: $E = mc^2$, and masses are quoted in $\text{MeV}/c^2$ with $1\ \text{u} = 931.5\ \text{MeV}/c^2$. So $m_e = 0.511$, $m_p = 938.28$, $m_n = 939.57\ \text{MeV}/c^2$.

Size and density of nuclei

• $R_X = R_0A^{1/3}$ with $R_0 = 1.2\times10^{-15}\ \text{m}$, so the nuclear volume is proportional to $A$.
• $\rho = \dfrac{3m}{4\pi R_0^3} = 2.3\times10^{17}\ \text{kg m}^{-3}$: independent of $A$, the same for every nucleus, and thirteen orders of magnitude greater than osmium, the densest element at $2.2\times10^{4}\ \text{kg m}^{-3}$.

Nuclear forces

• The strong (nuclear) force acts between nucleons and is mostly attractive; it is what holds the nucleus together against the electrostatic repulsion of the protons.
• Three properties the board wants: (i) it is the strongest of the subatomic forces, some $50$ to $60$ times stronger than the electrostatic force; (ii) its range is only a few fm, unlike the infinite range of electrostatic and gravitational forces, and it is negligible beyond that; (iii) it is charge independent — the same between $n$–$n$, $p$–$p$ and $n$–$p$ at the same separation.

15.8 Nuclear binding energy

• Mass defect: the difference between the actual mass of a nucleus and the sum of the masses of its constituent nucleons, $\Delta M = Zm_p + Nm_n - M$. The nucleus is always lighter than its parts.
• Binding energy: the energy that must be supplied to separate all the nucleons and take them far apart, $E_B = \Delta M c^2$; the same energy is released when the nucleus is assembled from free nucleons.
• With atomic-mass data use $E_B = \left(Zm_H + Nm_n - {}^{A}_{Z}M\right)c^2$ — the $Z$ electron masses added and subtracted cancel.
• Binding energy per nucleon $\dfrac{E_B}{A}$ is the average energy needed to free one nucleon and is the measure of stability: the larger it is, the more stable the nucleus.
• Shape of the $\dfrac{E_B}{A}$ against $A$ curve, to describe in words: it starts very low, deuterium having the smallest value and being the least stable nucleus; it rises steeply, flattens into a plateau for $A$ between $50$ and $80$ where the nuclei are most stable, peaks near $A = 56$ (iron), and then falls gradually for $A > 80$.
• The binding energy of the hydrogen nucleus, a single proton, is zero.
• Worked values worth remembering: alpha particle $28.29\ \text{MeV}$ ($7.07\ \text{MeV}$ per nucleon), ${}^{7}_{3}\text{Li}$ about $39.2\ \text{MeV}$, americium-244 about $7.5\ \text{MeV}$ per nucleon.

15.9 Radioactive decays

• Unstable nuclei change into other nuclei by emitting particles. The decaying nucleus is the parent, the one produced is the daughter, and the process is radioactive decay, discovered by Becquerel; the term radioactivity was coined by Marie Curie.

• Alpha decay happens in all nuclei with $A > 210$: too many protons, electrostatic repulsion beats the nuclear attraction, so the nucleus sheds protons two at a time. Example: ${}^{212}_{83}\text{Bi} \longrightarrow {}^{208}_{81}\text{Tl} + \alpha$.
• Beta-minus decay converts a neutron into a proton, $n \to p + e^- + \bar\nu$. Example: ${}^{60}_{27}\text{Co} \longrightarrow {}^{60}_{28}\text{Ni} + e^- + \bar\nu$.
• Beta-plus decay converts a proton into a neutron, $p \to n + e^+ + \nu$. Since the neutron is heavier, this cannot happen to a free proton; inside a nucleus the extra energy comes from the rest of the nucleus. Example: ${}^{22}_{11}\text{Na} \longrightarrow {}^{22}_{10}\text{Ne} + e^+ + \nu$.
• Gamma decay follows an alpha or beta decay that leaves the daughter in an excited nuclear state. Example: ${}^{57}\text{Co}$ beta-plus decays to ${}^{57}\text{Fe}$ in a state $136\ \text{keV}$ above the ground state, which de-excites either by one $136\ \text{keV}$ photon or by a $122\ \text{keV}$ photon followed by a $14\ \text{keV}$ photon.
• Atomic energy levels are separated by a few eV, nuclear levels by keV to MeV — hence atoms radiate from ultraviolet to radio, nuclei radiate gamma rays.
• The daughter is often unstable too, giving a decay series; ${}^{238}\text{U}$ takes fourteen alpha and beta decays to reach stable ${}^{206}\text{Pb}$.
• The $Q$-value equals the total kinetic energy of the products, so it is also the maximum kinetic energy of the beta particle (the case where the neutrino gets nothing).

15.10 Law of radioactive decay

• Statement: the number of nuclei undergoing decay per unit time is proportional to the number of undecayed nuclei present at that instant.
• $dN = -\lambda N(t)dt$, integrating to $N(t) = N_0e^{-\lambda t}$; $\lambda$ is the decay constant and the minus sign says $N$ falls with time.
• Activity $A(t) = -\dfrac{dN}{dt} = \lambda N(t) = A_0e^{-\lambda t}$, measured in becquerel ($1$ decay per second) or curie ($1\ \text{Ci} = 3.7\times10^{10}\ \text{Bq}$).
• Half-life $T_{1/2} = \dfrac{0.693}{\lambda}$: putting $N = \dfrac{N_0}{2}$ in the decay law. After each further half-life the number halves again — $N_0 \to \dfrac{N_0}{2} \to \dfrac{N_0}{4} \to \dfrac{N_0}{8}$ — and never reaches zero.
• Average (mean) life $\tau = \dfrac{1}{\lambda}$, found by averaging $t$ over $\lambda N_0e^{-\lambda t}dt$; hence $T_{1/2} = 0.693\,\tau$, so $\tau > T_{1/2}$.
• Which nucleus decays and when is unpredictable; only the statistical behaviour of a large number is described by the law.
• Other exponential decays for comparison: amplitude of a damped pendulum $A = A_0e^{-bt}$; Newton cooling $T = T_0e^{-kt}$; discharge of a capacitor $Q = Q_0e^{-t/RC}$.

15.11 Nuclear energy

• Chemical reactions release a few eV per reaction, nuclear reactions a few MeV — about a million times more energy per unit mass of fuel.
• Both routes to nuclear energy exploit the same fact: the products have a higher binding energy per nucleon than the reactants, so they are lighter, and the missing mass appears as energy, $Q = (\Delta M)c^2$.

Nuclear fission

• Splitting a heavy nucleus into two lighter nuclei with release of energy. Middle-weight nuclei ($A$ from $50$ to $80$) have the highest $\dfrac{E_B}{A}$, so a nucleus of $A \approx 230$ breaking into two fragments of $A$ between $50$ and $150$ releases energy.
• ${}^{236}_{92}\text{U}$ is the working fuel nucleus. It is not found in nature; it is made by letting naturally occurring ${}^{235}_{92}\text{U}$ absorb a slow neutron. Natural uranium is more than $99\%$ ${}^{238}_{92}\text{U}$ and less than $1\%$ ${}^{235}_{92}\text{U}$.
• Three textbook fission channels of ${}^{236}_{92}\text{U}$: to ${}^{137}_{53}\text{I} + {}^{97}_{39}\text{Y} + 2n$; to ${}^{140}_{56}\text{Ba} + {}^{94}_{36}\text{Kr} + 2n$; to ${}^{133}_{51}\text{Sb} + {}^{99}_{41}\text{Nb} + 4n$. About $177$ to $200\ \text{MeV}$ is released, appearing as kinetic energy of the products, i.e. as heat.
• Chain reaction: each fission releases on average $2.7$ neutrons, which are absorbed by further ${}^{235}\text{U}$ nuclei; unchecked, the number of fissions cascades and the result is an explosion.
• Nuclear reactor: a device in which fission is carried out in a controlled way, the chain reaction being prevented from running away; the heat produced is carried off and converted to electricity with turbines. India has 22 reactors, the largest at Kudankulam in Tamil Nadu; the USA generates the most nuclear power.