12.1 Discovery of the Neutron12.2 Nuclear Properties12.3 The Deuteron12.4 Nuclear Forces12.5 Nuclear Stability12.6 Radioactive Decay12.7 Alpha, Beta, and Gamma Decay12.8 Radioactive Nuclides

CHAPTER 12The Atomic Nucleus

It is said that Cockroft and Walton were interested in raising the voltage of their equipment, its reliability, and so on, more and more, as so often happens when you are involved with technical problems, and that eventually Rutherford lost patience and said, “If you don’t put a scintillation screen in and look for alpha particles by the end of the week, I’ll sack the lot of you.” And they went and found them (the first nuclear transmutations).

- Sir Rudolf Peierls in Nuclear Physics in Retrospect

12.1: Discovery of the Neutron

Rutherford proposed the atomic structure with the massive nucleus in 1911. Scientists knew which particles compose the nucleus in 1932.

Three (3) reasons why electrons cannot exist within the nucleus:

1) Nuclear sizeThe uncertainty principle puts a lower limit on its kinetic energy that is much larger that any kinetic energy observed for an electron emitted from nuclei.

2) Nuclear spinIf a deuteron (mass number A = 2 and atomic number Z = 1) consists of protons and electrons, the deuteron must contain 2 protons and 1 electron in order to have A = 2 and Z = 1.A nucleus composed of 3 fermions must result in a half-integral spin. But, it has been measured to be 1.

3) Nuclear magnetic moment:The magnetic moment of an electron is over 1000 times larger than that of a proton.The measured nuclear magnetic moments are on the same order of magnitude as the proton’s, so an electron is not a part of the nucleus.

Discovery of the Neutron In 1930 the German physicists Bothe and Becker used a radioactive polonium source

that emitted α particles. When these α particles bombarded beryllium, the radiation penetrated several centimeters of lead.

Photons are called gamma rays when they originate from the nucleus. They have energies on the order of MeV (as compared to X-ray photons due to electron transitions in atoms with energies on the order of KeV.)

Curie and Joliot performed several measurements to study penetrating high-energy gamma rays.

In 1932 Chadwick proposed that the new radiation produced by α + Be consisted of neutrons. His experimental data estimated the neutron’s mass as somewhere between 1.005 u and 1.008 u, not far from the modern value of 1.0087 u.

12.2: Nuclear Properties The nuclear charge is +e times the number (Z) of protons.

Hydrogen’s isotopes:

The nuclei of the deuterium and tritium atoms are called deuterons and tritons. Atoms with the same Z, but different mass number A, are called isotopes.

The symbol of an atomic nucleus is .Z = atomic number (number of protons)N = neutron number (number of neutrons)A = mass number (Z + N)X = chemical element symbol

Each nuclear species with a given Z and A is called a nuclide (핵종). Z characterizes a chemical element. The dependence of the chemical properties on N is negligible. Nuclides with the same neutron number are called isotones and the same value of A

are called isobars.

Isotropes NA Half-life Name1H 99.985% Stable 1 proton + 0 neutron Protium2H 0.015% stable 1 proton + 1 neutron Deuterium3H Very rare Radiative

(12.32 y)1 proton + 2 neutron Tritium

Atomic masses are denoted by the symbol u. 1 u = 1.66054 × 10−27 kg = 931.49 MeV/c2

Both neutrons and protons, collectively called nucleons, are constructed of other particles called quarks.

Nuclear Properties

Sizes and Shapes of Nuclei Rutherford concluded that the range of the nuclear force must be less than about 10−14 m. Assume that nuclei are spheres of radius R. Particles (electrons, protons, neutrons, and alphas) scatter when projected close to the

nucleus. It is not obvious whether the maximum interaction distance refers to the nuclear size

(matter radius), or whether the nuclear force extends beyond the nuclear matter (force radius).

The nuclear force is often called the strong force. Nuclear force radius ≈ mass radius ≈ charge radius

The nuclear radius may be approximated to be R = r0A1/3 , where r0 ≈ 1.2 × 10−15 m.

We use the femtometer with 1 fm = 10−15 m, or the fermi. The lightest nuclei by the Fermi distribution for the nuclear

charge density ρ(r) is

Nuclear Density:If we approximate the nuclear shape as a sphere, then we have: The nuclear mass density can be determined from (Au/V) to be 2.3 x 1017 kg/m3.

The nucleus is about 1014 times denser than ordinary matter!

3 30

4 43 3

V R r A

Nuclear Density, Intrinsic Spin, Intrinsic Magnetic Moment

Intrinsic Spin:The neutron and proton are fermions with spin quantum numbers s = ½. The spin quantum numbers are those previously learned for the electron.

Nuclear magnetic moments are measured in units of the nuclear magneton μN.

The divisor in calculating μN is the proton mass mp, which makes the nuclear magnetonsome 1800 times smaller than the Bohr magneton.

The proton magnetic moment is μp = 2.79μN.(Note the magnetic moment of the electron is μe = −1.00116μB.)

The neutron magnetic moment is μn = −1.91μN.The nonzero neutron magnetic moment implies that the neutron has negative and positiveinternal charge components at different radii Complex internal charge distribution.

Intrinsic Magnetic Moment:

Nuclear Magnetic Resonance (NMR)NMR results from resonant absorption of electromagnetic energy by a nucleus (mostly protons, 1H hydrogen atoms) changing its spin orientation

The resonance frequency depends on the chemical environment of the nucleus giving a specific finger print of particular groups (NMR spectroscopy)

NMR is nondestructive and contact free

Modern variants of NMR provide 3D structural resolution of proteins in solution

NMR tomography (Magnetic resonance imaging, MRI) is the most advanced and powerful imaging tool

12.3: The Deuteron ((Nucleus of Deuterium)

How the neutron and proton are bound together in a deuteron: The deuteron mass = 2.013553 u The mass of a deuteron atom = 2.014102 u The difference = 0.000549 u; the mass of an electron The deuteron nucleus is bound by an energy Bd, which represents mass-energy.

The mass of a deuteron is Add an electron mass

After the proton, the next simplest nucleus is the deuteron (symbol D or 2H, also known as heavy hydrogen) is one of two stable isotopes of hydrogen. Deuteron = one proton + one neutron (0.0156% of all the natural hydrogen in the oceans)

md + me is the atomic deuterium mass M(2H); mp + me is the atomic hydrogen mass.

The binding energy of any nucleus, the energy required to separate the nucleus into free neutrons and protons.

Nuclear binding energy The binding energy of any nucleus

Experimental verification of Deuteron’s Binding Energies We can check the result for the 2.22-MeV binding energy of deuteron by using a nuclear reaction.

Incidence of gamma rays to deuteron gas breaks up a deuteron into a neutron and a proton:

This nuclear reaction is called photodisintegration or a photonuclear reaction. The mass-energy relation is

where hf is the incident photon energy, Kn and Kp are the neutron and proton kinetic energies.

For the minimum energy required for the photodisintegration, we let Kn = Kp = 0.

It is not quite correct, because momentum must also be conservedin the reaction (Kn and Kp can’t both be zero). The precise relation is

Deuteron Spin and Magnetic MomentDeuteron’s nuclear spin quantum number is 1 neutron and proton spins are aligned parallel to each other.The nuclear magnetic moment of a deuteron is 0.86μN Sum of free proton and neutron 2.79μN − 1.91μN = 0.88μN.

12.4: Nuclear ForcesTo study nuclear forces, the most straightforward are based on scattering experiments. (Neutron + proton), (proton + proton) elastic scatterings reveals the nuclear potential.

Nuclear potential

~3 fm

Attractive force in short range(~ 40 MeV in depth)

Nuclear forces are charge independent!

Coulomb potential

12.5: Nuclear Stability A nucleus containing A nucleons is said to be stable

if its mass is smaller than that of any other possible combination of A nucleons Binding energy is positive.

Unstable nuclei

For A ≤ 40, nature prefers the number of protons and neutrons in the nucleus to be about the same Z ≈ N.

However, for A ≥ 40, there is a decided preference for N > Z. As the number (Z) of protons increases, the Coulomb force

between all the protons becomes stronger until it eventually affects the binding significantly.

For heavy nuclei, the nucleus will have a preference for fewer protons than neutrons because of the large Coulomb repulsion energy.

There are no stable nuclei with Z > 83 because of the increasingly larger Coulomb force.

The heaviest known stable nucleus is 83Bi. All nuclei with Z > 83 and A > 209 will eventually decay

spontaneously into some combination of smaller masses.

Most stable nuclides have both even Z and even N(even-even nuclides).

Only four stable nuclides have odd Z and odd N(odd-odd nuclides):

Z = atomic number (number of protons)N = neutron number (number of neutrons)A = mass number (Z + N)X = chemical element symbol

Stable nuclei

The Liquid Drop Model

Niels Bohr, Carl F. von Weizsacker, and others in the 1930s They treats the nucleus as a collection of interacting particles in a liquid drop. The total binding energy, the semi-empirical mass formula is

The volume term (av) indicates that the binding energy is approximately the sum of all the interactions between the nucleons. The second term is called the surface effect because the nucleons on the nuclear surface are not completely surrounded by

other nucleons. The third term is the Coulomb energy The fourth term is due to the symmetry energy. In the absence of Coulomb forces, the nucleus prefers to have N ≈ Z and

has a quantum-mechanical origin, depending on the exclusion principle. The last term is due to the pairing energy and reflects the fact that the nucleus is more stable for even-even nuclides.

No nuclide heavier than has been found in nature. If they ever existed, they must have decayed so quickly that quantities sufficient to measure no longer exist.

Binding Energy Per Nucleon It peaks near A = 56 The curve increases rapidly, demonstrating the saturation effect of nuclear force Sharp peaks for the even-even nuclides 4He, 12C, and 16O

12.6: Radioactive Decay The discoverers of radioactivity were Wilhelm Röntgen, Henri Becquerel, Marie Curie

and her husband Pierre in the late 1890s. Marie Curie and her husband Pierre discovered polonium and radium in 1898.

The simplest decay form is that of a gamma ray ( decay), which represents the nucleus changing from an excited state to lower energy state. (No change in N or Z)

Other modes of decay include emission of α particles ( decay), β particles ( decay), protons, neutrons, and fission.

Activity = the disintegrations or decays per unit time for N unstable atoms in a material:

where we insert – sign since dN / dt is negative because total number N decreases with time.

SI unit of activity is the becquerel: 1 Bq = 1 decay / s For example, natural potassium (40K) in a typical human body produces 4,000 disintegrations per second, 4 kBq of activity.

The global inventory of carbon-14 is estimated to be 8.5×1018 Bq (8.5 EBq, 8.5 exabecquerel). The nuclear explosion in Hiroshima (14 kt or 59 TJ) is estimated to have produced 8×1024 Bq (8 YBq, 8 yottabecquerel).

The curie (symbol Ci) is a non-SI unit of radioactivity: 1 Ci = 3.7 × 1010 Bq = 37 GBq

(Note) The sievert (symbol: Sv) is the SI unit of radiation dose for considering biological effects. One sievert carries with it a 5.5% chance of eventually developing cancer.Doses greater than 1 sievert received over a short time period, possibly leading to death within weeks.

When a nucleus decays, all the conservation laws must be observed: Mass-energy, Linear momentum, Angular momentum, Electric charge And, the total number of nucleons Conservation of nucleons

The total number of nucleons (A, the mass number) must be conserved in a low-energy nuclear reaction (say, less than 938 MeV) or decay. (At higher energies enough rest energy may be available to create nucleons)

12.7: Alpha, Beta, and Gamma Decay

Let the radioactive nucleus having the mass decay to products: lighter-one mass My and heavier-one MD. The conservation of energy is

where Q is the energy released (disintegration energy), the negative of the binding energy B.

If B > 0, a nuclide is bound and stable If Q > 0 (B < 0), a nuclide is unbound, unstable, and may decay.

In natural, decays emitting nucleons do not occur because the masses are such that Q < 0.

Alpha Decay The nucleus 4He is very stable, having a binding energy of 28.3 MeV. The combination of two neutrons and two protons is particularly strong

because of the pairing effects. If the last two protons and two neutrons in a large nucleus are bound

by less than 28.3 MeV, then the emission of an alpha particle (alpha decay) is possible.

If Q > 0, alpha decay is possible.

(Example) Consider the Uranium nucleus.

The alpha particle is the same as the nucleus of a helium-4 atom (two protons and two neutrons)It also has a charge +2e.

Alpha decay is allowed, because Q > 0.

Radioactive decay in which an alpha particle (two protons and two neutrons) is emitted from an atomic nucleus.

Beta Decay Unstable nuclei may move closer

to the line of stability by undergoing beta decay.

The decay of a free neutron is

The beta decay of 14C (unstable) to form 14N, a stable nucleus, can be written as

Radioactive decay in which a beta particle (an electron or a positron) is emitted from an atomic nucleus.

(electron neutrino)

There was a problem in neutron decay: the spin ½ neutron cannot decay to two spin ½ particles, a proton and an electron.

Wolfgang Pauli suggested a neutrino that must be produced in beta decay. It has spin quantum number ½, charge 0, and carries away the additional energy.

Neutrino has little or no mass, its energy may be all kinetic. Neutrinos have no charge and do not interact electromagnetically. They are not affected by the strong force of the nucleus.

Beta decay is the reflection of a special kind of force weak force The electromagnetic force and weak forces are two types of the electroweak force.

β- Decay

In the general,

The weak interaction converts an atomic nucleus into a nucleus with one higher atomic number while emitting an electron (e−) and an electron antineutrino (e):

β+ Decay Positron emission: the weak interaction converts a nucleus into its next-lower neighbor on the periodic table while emitting a positron (e+) and an electron neutrino ( ):

In the general,

Electron Capture: one other form of β Decay Classically, inner K-shell and L-shell electrons are tightly bound and their orbits are highly elliptical,

these electrons spend a time passing through the nucleus, thereby the possibility of atomic electron capture.

The general reaction is,

In all cases where β+ decay of a nucleus is allowed energetically, the electron capture process is also allowed.

(or, K-capture)

Gamma Decay Radioactive decay in which electromagnetic radiation of extremely high frequency and therefore high energy per photon emits from an atomic nucleus.

The gamma-ray energy hf is given by the difference of the higher energy state E> and lower one E<.

A transition between two nuclear excited states E> and E< is

Gamma rays from radioactive decay commonly have energies of a few hundred keV, and almost always less than 10 MeV.

By contrast, the energies of gamma rays from astronomical sources can be much higher, ranging over 10 TeV, at a level far too large to result from radioactive decay.[

The effect of gamma and other ionizing radiation on living tissue is more closely related to the amount of energy deposited This is called the absorbed dose.

The sievert (Sv) is the SI unit of absorbed dose: the amount of radiation required to deposit 1 joule of energy in 1 kilogram of any kind of matter.

12.8: Radioactive Nuclides The unstable nuclei found in nature exhibit natural radioactivity.

Radioactive Nuclides The radioactive nuclides made in the laboratory exhibit artificial radioactivity. Heavy radioactive nuclides can change their mass number only by alpha

decay (AX → A−4D) but can change their charge number Z by either alpha or beta decay.

There are only four paths that the heavy naturally occurring radioactive nuclides may take as they decay.

Mass numbers expressed by either: 4n 4n + 1 4n + 2 4n + 3

Radioactive Nuclides The sequence of one of the radioactive series 232Th

212Bi can decay by either alpha or beta decay (branching).

Time Dating Using Lead Isotopes A plot of the abundance ratio of 206Pb / 204Pb versus 207Pb / 204Pb can be a

sensitive indicator of the age of lead ores. Such techniques have been used to show that meteorites, believed to be left

over from the formation of the solar system, are 4.55 billion years old. The growth curve for lead ores from various deposits:

The age of the specimens can be obtained from the abundance ratio of 206Pb/204Pb versus 207Pb/204Pb.

Radioactive Carbon Dating

Radioactive 14C is produced in our atmosphere by the bombardment of 14N by neutrons produced by cosmic rays.

When living organisms die, their intake of 14C ceases, and the ratio of 14C / 12C (= R) decreases as 14C decays.

The period just before 9000 years ago had a higher 14C / 12C ratio by factor of about 1.5 than it does today.

Because the half-life of 14C is 5730 years, it is convenient to use the 14C / 12C ratio to determine the age of objects over a range up to 45,000 years ago.