Chapter 44

Nuclear Structure

Milestones in the Development of Nuclear Physics

1896: the birth of nuclear physics Becquerel discovered radioactivity in uranium

compounds Rutherford showed the radiation had three

types: alpha (He nuclei) beta (electrons) gamma (high-energy photons)

More Milestones

1911 Rutherford, Geiger and Marsden performed scattering experiments Established that the nucleus could be treated as a

point mass and a point charge Most of the atomic mass was contained in the

nucleus Nuclear force was a new type of force

Some Properties of Nuclei

All nuclei are composed of protons and neutrons Exception is ordinary hydrogen with a single

proton The atomic number Z equals the number of

protons in the nucleus Sometimes called the charge number

The neutron number N is the number of neutrons in the nucleus

More Properties of Nuclei

The mass number A is the number of nucleons in the nucleus A = Z + N Nucleon is a generic term used to refer to either a

proton or a neutron The mass number is not the same as the mass

Symbolism

X is the chemical symbol of the element Example:

Mass number is 27 Atomic number is 13 Contains 13 protons Contains 14 (27 – 13) neutrons

The Z may be omitted since the element can be used to determine Z

XAZ

Al2713

More Properties

The nuclei of all atoms of a particular element must contain the same number of protons

They may contain varying numbers of neutrons Isotopes of an element have the same Z but

differing N and A values The natural abundance of isotopes can vary Isotope example:

11 12 13 146 6 6 6C C C C, , ,

Charge

The proton has a single positive charge, e The electron has a single negative charge,

- e The neutron has no charge

Made it difficult to detect in early experiments Easy to detect with modern devices

e = 1.602 177 33 x 10-19 C

Mass

It is convenient to use atomic mass units, u, to express masses 1 u = 1.660 539 x 10-27 kg Based on definition that the mass of one atom of

12C is exactly 12 u Mass can also be expressed in MeV/c2

From ER = mc2

1 u = 931.494 MeV/c2

Includes conversion 1 eV = 1.602 177 x 10-19 J

Some Masses in Various Units

The Size of the Nucleus

First investigated by Rutherford in scattering experiments He found an expression for how close an alpha particle

moving toward the nucleus can come before being turned around by the Coulomb force

From conservation of energy, the kinetic energy of the particle must be completely converted to potential energy

Active Figure 44.1

Use the active figure to adjust the atomic number of the target nucleus

Also adjust the kinetic energy of the alpha particle

Observe the approach of the alpha particle

PLAYACTIVE FIGURE

Size of the Nucleus, cont.

d is called the distance of closest approach d gives an upper limit for the size of the nucleus

Rutherford determined that

For gold, he found d = 3.2 x 10-14 m

2

24 e

Zed k

mv

More About Size

Rutherford concluded that the positive charge of the atom was concentrated in a sphere whose radius was no larger than about 10-14 m He called this sphere the nucleus

These small lengths are often expressed in femtometers (fm) where 1 fm = 10-15 m Also called a fermi

Size of Nucleus, Final

Since the time of Rutherford, many other experiments have concluded the following: Most nuclei are approximately spherical Average radius is

ro = 1.2 x 10-15 m A is the mass number

1 3or r A

Density of Nuclei The volume of the nucleus

(assumed to be spherical) is directly proportional to the total number of nucleons

This suggests that all nuclei have nearly the same density

Nucleons combine to form a nucleus as though they were tightly packed spheres

Nuclear Stability

There are very large repulsive electrostatic forces between protons These forces should cause the nucleus to fly apart

The nuclei are stable because of the presence of another, short-range force, called the nuclear force This is an attractive force that acts between all nuclear

particles The nuclear attractive force is stronger than the

Coulomb repulsive force at the short ranges within the nucleus

Features of the Nuclear Force

Attractive force that acts between all nuclear particles

Very short range It falls to zero when the separation between

particles exceeds about several fermis Independent of charge

The nuclear force on p-p, p-n, n-n are all the same

Does not affect electrons

Nuclear Stability, cont.

Light nuclei are most stable if N = Z

Heavy nuclei are most stable when N > Z Above about Z = 20 As the number of protons

increases, the Coulomb force increases and so more neutrons are needed to keep the nucleus stable

No nuclei are stable when Z > 83

Binding Energy

The total energy of the bound system (the nucleus) is less than the combined energy of the separated nucleons This difference in energy is called the binding

energy of the nucleus It can be thought of as the amount of energy you need

to add to the nucleus to break it apart into its components

Binding Energy, cont.

The binding energy can be calculated from conservation of energy and the Einstein mass-energy equivalence principle:

Eb (MeV) = [ZM(H) + Nmn – M (AZX)] x

931.494 MeV/u M(H) is the atomic mass of the neutral hydrogen atom M (A

ZX) represents the atomic mass of an atom of the isotope (A

ZX) Mn is the mass of the neutron

The masses are expressed in atomic mass units

Binding Energy per Nucleon

Notes from the Graph

The curve peaks in the vicinity of A = 60 Nuclei with mass numbers greater than or less than

60 are not as strongly bound as those near the middle of the periodic table

There is a decrease in binding energy per nucleon for A > 60 Energy is released when a heavy nucleus splits or

fissions Energy is released since each product nucleus are more

tightly bound to one another than are the nucleons of the original nucleus

More Notes from the Graph

The binding energy is about 8 MeV per nucleon for nuclei with A > 50 This suggests that the nuclear force

saturates A particular nucleon can interact with only a

limited number of other nucleons has the largest binding energy per

nucleon

6228Ni

Nuclear Models

Two models of the nucleus will be discussed Liquid-drop model

Provides good agreement with observed nuclear binding energies

Shell model Predicts the existence of stable nuclei

Liquid-Drop Model

Nucleons are treated like molecules in a drop of liquid

The nucleons interact strongly with one another

They undergo frequent collisions as they jiggle around in the nucleus

The jiggling motion is analogous to the thermally agitated motion of molecules in a drop of liquid

Liquid-Drop Model – Effects Influencing Binding Energy, 1

The volume effect The nuclear force on a given nucleon is due only

to a few nearest neighbors and not to all the other nucleons in the nucleus

The total binding energy is proportional to A and therefore proportional to the nuclear volume

This contribution to the binding energy of the entire nucleus is C1A C1 is an adjustable constant

Liquid-Drop Model – Binding Energy Effect 2

The surface effect Nucleons on the surface have fewer neighbors

than those in the interior Surface nucleons reduce the binding energy by

an amount proportional to their number The number of nucleons is proportional to the

surface area The surface term can be expressed as –C2A2/3

C2 is a second adjustable constant

Liquid-Drop Model – Binding Energy Effect 3

The Coulomb repulsion effect Each proton repels every other proton in the

nucleus The potential energy associated with the Coulomb

force is proportional to the number of protons, Z The reduction in the binding energy due to the

Coulomb effect is –C3Z(Z - 1)/A1/3

C3 is another adjustable constant

Liquid-Drop Model – Binding Energy Effect 4

The symmetry effect Any large symmetry between N and Z for light nuclei

reduces the binding energy For larger A, the value of N for stable nuclei is larger The effect can be described by a binding energy term in

the form –C4(N - Z)2 / A For small A, any large asymmetry between N and Z makes

the term large For large A, the A in the denominator reduces the value of

the term so that it has little effect on the overall binding energy

C4 is another adjustable constant

Liquid-Drop Model – Binding Energy Effect Summary

Putting these terms together results in the semiempirical binding-energy formula:

The four constants are adjusted to fit the theoretical expression to the experimental data For A 15, C1 = 15.7 MeV; C2 = 17.8 MeV; C3 =

0.71 MeV; and C4 = 23.6 MeV

2

2 31 2 3 41 3

1b

Z Z N ZE C A C A C C

A A

Liquid Drop Model, Final

The equation fits the known nuclear mass values very well

Does not account for some of the finer details of nuclear structure Stability Angular momentum

Features of Binding Energy

When binding energies are studied closely it is found that: Most stable nuclei have an even value of A

Only 8 stable nuclei have odd values for both A and Z There is a difference between the binding energy

per nucleon given by the semiempirical formula and experiments

Features of Binding Energy – Magic Numbers

The disagreement between the semiempirical formula and experiments is plotted

Peaks appear in the graph These peaks are at the magic numbers of

Z or N = 2, 8, 20, 28, 52, 82

Features of Binding Energy, cont.

Studies of nuclear radii show deviations from the expected values Graphs of the data show peaks at values of N

equal to the magic numbers A group of isotones is a collection of nuclei

having the same value of N and different values of Z When the number of stable isotones is graphed as

a function of N, there are peaks at the magic numbers

Features of Binding Energy, final

Several other nuclear measurements show anomalous behavior at the magic numbers

The peaks are reminiscent of the peaks in graphs of ionization energy of atoms and lead to the shell model of the nucleus

Maria Goeppert-Mayer

1906 – 1972 German scientist Best known for her

development of the shell model of the nucleus

Shared the Nobel Prize in 1963 Shared with Hans Jensen

who simultaneously developed a similar model

Shell Model

The shell model is also called the independent-particle model

In this model, each nucleon is assumed to exist in a shell Similar to atomic shells for electrons

The nucleons exist in quantized energy states There are few collisions between nucleons

Shell Model, cont.

Each state can contain only two protons or two neutrons They must have opposite

spins They have spins of ½, so

the exclusion principle applies

The set of allowed states for the protons differs from the set of allowed states for the neutrons

Shell Model, final

Proton energy levels are farther apart than those for neutrons due to the superposition of the Coulomb force and the nuclear force for the protons

The spin-orbit effect for nucleons is due to the nuclear force The spin-orbit effect influences the observed

characteristics of the nucleus

Shell Model Explanation of Experimental Results

Nuclei with even numbers of protons and neutrons are more stable Any particular state is filled when it contains two

protons or two neutrons An extra proton or neutron can be added only at

the expense of increasing the nucleus’s energy This increase in energy leads to greater instability

in the nucleus

Shell Model Explanation of Experimental Results, cont.

Nuclei tend to have more neutrons than protons Proton energy levels are higher As Z increases and higher states are filled, a proton level

for a given quantum number will be much higher in energy than the neutron level for the same quantum number

It is more energetically favorable for the nucleus to form with neutrons in the lower energy levels than protons in the higher levels

So, the number of neutrons is greater than the number of protons

Marie Curie

1867 – 1934 Polish scientist Shared Nobel Prize in 1903

for studies in radioactive substances Prize in physics Shared with Pierre Curie

and Becquerel Won Nobel Prize in 1911 for

discovery of radium and polonium Prize in chemistry

Radioactivity

Radioactivity is the spontaneous emission of radiation Discovered by Becquerel in 1896 Many experiments were conducted by Becquerel

and the Curies Experiments suggested that radioactivity was

the result of the decay, or disintegration, of unstable nuclei

Radioactivity – Types

Three types of radiation can be emitted Alpha particles

The particles are 4He nuclei Beta particles

The particles are either electrons or positrons A positron is the antiparticle of the electron It is similar to the electron except its charge is +e

Gamma rays The “rays” are high energy photons

Distinguishing Types of Radiation

The gamma particles carry no charge

The alpha particles are deflected upward

The beta particles are deflected downward A positron would be

deflected upward, but would follow a different trajectory than the α due to its mass

Penetrating Ability of Particles

Alpha particles Barely penetrate a piece of paper

Beta particles Can penetrate a few mm of aluminum

Gamma rays Can penetrate several cm of lead

The Decay Constant

The number of particles that decay in a given time is proportional to the total number of particles in a radioactive sample

λ is called the decay constant and determines the rate at which the material will decay

N is the number of undecayed radioactive nuclei present

No is the number of undecayed nuclei at time t = 0

gives λto

dNλN N N e

dt

Decay Curve

The decay curve follows the equation N = Noe-λt

The half-life is also a useful parameter The half-life is defined as

the time interval during which half of a given number of radioactive nuclei decay

1 2

ln 2 0693.T

λ λ

Active Figure 44.9

Use the active figure to adjust the half-life

Observe the decay curve

PLAYACTIVE FIGURE

Decay Rate

The decay rate R of a sample is defined as the number of decays per second

Ro = Noλ is the decay rate at t = 0 The decay rate is often referred to as the activity

of the sample

λto

dNR λN R e

dt

Units

The unit of activity, R, is the curie (Ci) 1 Ci ≡ 3.7 x 1010 decays/s

The SI unit of activity is the becquerel (Bq) 1 Bq ≡ 1 decay/s

Therefore, 1 Ci = 3.7 x 1010 Bq

The most commonly used units of activity are the millicurie and the microcurie

Decay Processes The blue circles are the stable

nuclei seen before Above the line the nuclei are

neutron rich and undergo beta decay (red)

Just below the line are proton rich nuclei that undergo beta (positron) emission or electron capture (green)

Farther below the line the nuclei are very proton rich and undergo alpha decay (yellow)

Active Figure 44.10

Click on any colored dot

Study the decay modes and decay energies

PLAYACTIVE FIGURE

Alpha Decay

When a nucleus emits an alpha particle it loses two protons and two neutrons N decreases by 2 Z decreases by 2 A decreases by 4

Symbolically X is called the parent nucleus Y is called the daughter nucleus

4 42 2X Y HeA A

Z Z

Decay – General Rules

The sum of the mass numbers A must be the same on both sides of the equation

The sum of the atomic numbers Z must be the same on both sides of the equation

When one element changes into another element, the process is called spontaneous decay or transmutation

Relativistic energy and momentum of the isolated parent nucleus must be conserved

Disintegration Energy

The disintegration energy Q of a system is defined as

Q = (Mx – My – Mα)c2

The disintegration energy appears in the form of kinetic energy in the daughter nucleus and the alpha particle

It is sometimes referred to as the Q value of the nuclear decay

Alpha Decay, Example

Decay of 226 Ra

If the parent is at rest before the decay, the total kinetic energy of the products is 4.87 MeV

In general, less massive particles carry off more of the kinetic energy

226 222 488 86 2Ra Rn He

Active Figure 44.11

Use the active figure to observe the decay of radium-226

PLAYACTIVE FIGURE

Alpha Decay, Notes

Experimental observations of alpha-particle energies show a number of discrete energies instead of a single value The daughter nucleus may be left in an excited

quantum state So, not all of the energy is available as kinetic energy

A negative Q value indicates that such a proposed decay does not occur spontaneously

Alpha Decay, Mechanism

In alpha decay, the alpha particle tunnels though a barrier

For higher energy particles, the barrier is narrower and the probability is higher for tunneling across This higher probability

translates into a shorter half-life of the parent

Beta Decay

During beta decay, the daughter nucleus has the same number of nucleons as the parent, but the atomic number is changed by one

Symbolically

Beta decay is not completely described by these equations

1

1

X Y e

X Y e

A AZ Z

A AZ Z

Beta Decay, cont.

The emission of the electron or positron is from the nucleus The nucleus contains protons and neutrons The process occurs when a neutron is

transformed into a proton or a proton changes into a neutron The electron or positron is created in the process of

the decay Energy must be conserved

Beta Decay – Particle Energy The energy released in the

decay process should almost all go to kinetic energy of the β particle Since the decaying nuclei

all have the same rest mass, the Q value should be the same for all decays

Experiments showed a range in the amount of kinetic energy of the emitted particles

Were conservation laws violated?

Neutrino

To account for this “missing” energy, in 1930 Pauli proposed the existence of another particle

Enrico Fermi later named this particle the neutrino Properties of the neutrino

Zero electrical charge Mass much smaller than the electron, probably not zero Spin of ½ Very weak interaction with matter and so is difficult to

detect

Beta Decay – Completed

Symbolically

is the symbol for the neutrino is the symbol for the antineutrino

To summarize, in beta decay, the following pairs of particles are emitted An electron and an antineutrino A positron and a neutrino

1

1

X Y e

X Y e

A AZ Z

A AZ Z

ν

ν

ν

Beta Decay – Examples

Active Figure 44.15

Use the active figure to observe the decay of Carbon-14

PLAYACTIVE FIGURE

Beta Decay, Final Notes

The fundamental process of e- decay is a neutron changing into a proton, an electron and an antineutrino

In e+, the proton changes into a neutron, positron and neutrino This can only occur within a nucleus It cannot occur for an isolated proton since its

mass is less than the mass of the neutron

Electron Capture

Electron capture is a process that competes with e+ decay

In this case, a parent nucleus captures one of its own orbital electrons and emits a neutrino:

In most cases, a K-shell electron is captured, so this is often referred to as K capture

01 1X e YA A

Z Z ν

Electron Capture, Detection

Because the neutrino is very hard to detect, electron capture is usually observed by the x-rays given off as higher-shell electrons cascade downward to fill the vacancy created in the K shell

Q Values for Beta Decay

For e- decay and electron capture, the Q value is Q = (Mx – MY)c2

For e+ decay, the Q value is

Q = (Mx – MY - 2me)c2

The extra term, -2mec2, is due to the fact that the atomic number of the parent decreases by one when the daughter is formed

To form a neutral atom, the daughter sheds one electron If Q is negative, the decay will not occur

Gamma Decay

Gamma rays are given off when an excited nucleus decays to a lower energy state

The decay occurs by emitting a high-energy photon called gamma-ray photons

The X* indicates a nucleus in an excited state Typical half-life is 10-10 s

X X*A AZ Z γ

Gamma Decay – Example

Example of a decay sequence The first decay is a beta emission The second step is a gamma emission

Gamma emission doesn’t change Z, N, or A The emitted photon has an energy of hƒ equal to

E between the two nuclear energy levels

12 125 6

12 126 6

B C e

C C

*

*

ν

γ

Summary of Decays

Natural Radioactivity

Classification of nuclei Unstable nuclei found in nature

Give rise to natural radioactivity Nuclei produced in the laboratory through nuclear reactions

Exhibit artificial radioactivity Three series of natural radioactivity exist

Uranium Actinium Thorium

Some radioactive isotopes are not part of any decay series

Radioactive Series, Overview

Decay Series of 232Th

Series starts with 232Th Processes through a

series of alpha and beta decays

The series branches at 212Bi

Ends with a stable isotope of lead, 208Pb

Nuclear Reactions

The structure of nuclei can be changed by bombarding them with energetic particles The changes are called nuclear reactions

As with nuclear decays, the atomic numbers and mass numbers must balance on both sides of the equation

Nuclear Reactions, cont.

A target nucleus, X, is bombarded by a particle a, resulting in a daughter nucleus Y and an outgoing particle b a + X Y + b

The reaction energy Q is defined as the total change in mass-energy resulting from the reaction Q = (Ma + MX – MY – Mb)c2

Q Values for Reactions

The Q value determines the type of reaction An exothermic reaction

There is a mass “loss” in the reaction There is a release of energy Q is positive

An endothermic reaction There is a “gain” of mass in the reaction Energy is needed, in the form of kinetic energy of the

incoming particles Q is negative The minimum energy necessary for the reaction to occur is

called the threshold energy

Nuclear Reactions, final

If a and b are identical, so that X and Y are also necessarily identical, the reaction is called a scattering event If the kinetic energy before the event is the same

as after, it is classified as elastic scattering If the kinetic energies before and after are not the

same, it is an inelastic scattering

Conservation Rules for Nuclear Reactions

The following must be conserved in any nuclear reaction Energy Momentum Total charge Total number of nucleons

Nuclear Magnetic Resonance (NMR)

A nucleus has spin angular momentum

Shown is a vector model giving possible orientations of the spin and its projection on the z axis

The magnitude of the spin angular momentum is

( 1)I I

NMR, cont.

For a nucleus with spin ½, there are only two allowed states Emax and Emin

It is possible to observe transitions between two spin states using NMR

MRI An MRI (Magnetic

Resonance Imaging) is based on NMR

Because of variations in an external field, hydrogen atoms in different parts of the body have different energy splittings between spin states

The resonance signal can provide information about the positions of the protons