Chapter 30

Nuclear Physics

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 nucleus) Beta (electrons) Gamma (high-energy photons)

More Milestones 1911 Rutherford, Geiger and

Marsden performed scattering experiments Established 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 just a

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 56 Atomic number is 26 Contains 26 protons Contains 30 (56-26) neutrons

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

XAZ

5626Fe

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 Example: CCCC 14

6

13

6

12

6

11

6

Charge The proton has a single positive

charge, +e The electron has a single negative

charge, -e The neutron has no charge

Makes it difficult to detect e = 1.60217733 x 10-19 C

Mass It is convenient to use atomic mass

units, u, to express masses 1 u = 1.660539 x 10-27 kg Based on definition that the mass of one

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

From ER = m c2

1 u = 931.494 MeV/c2

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

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 For silver, he found d = 2 x 10-14 m

2

2

4 ek Zed

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 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

31

oArr

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

Since r3 would be proportional to A

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 It is the strongest force in nature 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

Magic Numbers Most stable nuclei have even numbers

of A Certain values of N and Z correspond to

unusually high stability These values are called magic numbers

N or Z = 2, 8, 20, 28, 50, 82, 126 For example, He has N = 2 and Z = 2 and is

very stable

Nuclear Spin Protons and neutrons have intrinsic

angular momentum called spin A nucleus has a net intrinsic angular

momentum that arises from the individual spins of the protons and neutrons

This angular momentum must obey the same quantum rules as orbital angular momentum

Nuclear Angular Momentum The magnitude of the nuclear angular

momentum is due to the combination of all nucleons

It is equal to I is called the nuclear spin quantum

number It may be an integer or half-integer

1I I

Possible Orientations of Nuclear Spin Shown is a vector model

giving possible orientations of the spin and its projection on the z axis

This is for the case where I = 3/2

Nuclear Magneton The nuclear angular momentum has a nuclear

magnetic moment associated with it The magnetic moment is measured in terms of

the nuclear magneton, n

Note that n is smaller than B by a factor of about 2000, due to the difference in mass between the electron and the proton

275.05 102n

p

e JTm

Nuclear Magnetic Moment, final The magnetic moment of a free proton is

2.792 8n No general theory of nuclear magnetism explains

this value The neutron also has a magnetic moment,

even though it has no charge The magnetic moment of the neutron is –

1.913 5n The negative sign indicates the neutron’s magnetic

moment is opposite its spin angular momentum

Nuclear Magnetic Resonance (NMR) Because the direction of the magnetic

moment for a particle is quantized, the energies of the particle are also quantized

The spin vector cannot align exactly with the direction of the magnetic field This gives the extreme values of the

energy as ±zB z is the z component of the magnetic moment

NMR, cont These states are

often called spin states

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

The magnetic field splits the states

NMR, final The sample is irradiated with electromagnetic

waves in the radio range of the em spectrum The frequency of the radio waves is adjusted

so that the photon energy matches the separation energy between spin states

There is a net absorption of energy in the system which is detected by experimental control and measuring systems

MRI An MRI (Magnetic

Resonance Imaging) is based on NMR

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

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

Binding Energy The total rest energy of the bound

system (the nucleus) is less than the combined rest 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:

The masses are expressed in atomic mass units

M(H) is the atomic mass of the neutral hydrogen atom

mn is the mass of the neutron M(X) is the mass of that isotope

Ab n ZE (MeV) = [ZM(H) + Nm - M ( X)] x 931.494 MeV/u

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

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

Marie Curie 1867 – 1934 Shared Nobel Prize in

1903 for studies in radioactive substances

Prize in physics Shared with Pierre Curie

and Becquerel Won Noble 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 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 rate at which a decay process occurs is

proportional to the radioactive nuclei present in the 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

to

dNN gives N N e

dt

Decay Rate The decay rate, R, of a sample is

defined as the number of decays per second

Ro = No is the decay rate at t = o The decay rate is often referred to as the

activity of the sample

t to o

dNR N e R e

dt

Decay Curve The decay curve follows

the equation N = No e- 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

693.02lnT 21

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)

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

HeYX 42

4A2Z

AZ

Decay – General Rules When one element changes into another

element, the process is called spontaneous decay or transmutation

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

The total energy and total momentum of the system must be conserved in the decay

Disintegration Energy The disintegration energy, Q of a

system is defined as

Q = (Mx – My – M) 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

HeRnRa 42

22286

22688

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 through 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

eYX

eYXA1Z

AZ

A1Z

AZ

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 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

Neutrino To account for this “missing” energy and

momentum, 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 but

less than 2.8 eV/c2

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

eYX

eYXA1Z

AZ

A1Z

AZ

Beta Decay – Examples

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, a process often referred to as K capture

YeX A

1Z

0

1

A

Z

Carbon Dating Beta decay of C-14 is used in dating organic

materials

The process depends on the ratio of C-14 to C-12 in the atmosphere which is relatively constant

When an organism dies, the ratio decreases as a result of the beta decay of the C-14

14 146 7C N e

Enrico Fermi 1901 – 1954 Awarded Nobel Prize in

physics in 1938 for producing transunranic elements and discovery of nuclear reactions produced by slow neutrons

Other contributions include: Theory of beta decay Free electron theory of metals Development of first fission

reactor (1942)

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

The X* indicates a nucleus in an excited state

X*X A

Z

A

Z

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

C*C

e*CB126

126

126

125

Summary of Decays

Nuclear Reactions 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

Types of Nuclear Reactions One important feature of nuclear

reactions is that much more energy is released than with normal chemical reactions

Two types of reactions can occur Fission Fusion

Nuclear Fission A heavy nucleus splits into two smaller nuclei A fissionable nucleus (X) absorbs a slowly

moving neutron (a) and splits into two smaller nuclei (Y1 and Y2), releasing energy and multiple neutrons (several particles b)

With no means of control, a chain reaction explosion occurs

With controls, the fission process is used in nuclear power generating plants

Chain Reaction – Diagram

Nuclear Fusion Nuclear fusion occurs when two light nuclei

combine to form a heavier nucleus This is difficult since the nuclei must

overcome the Coulomb repulsion before they are close enough to fuse One way to do this is to cause them to move with

high kinetic energy by raising them to a high temperature

Also need a high density

Fusion in the Sun The centers of stars have high enough

temperatures and densities to generate energy through fusion

The Sun fuses hydrogen Some stars fuse heavier elements

Reactions in cool stars (T < 15 x 106 K) take place through the proton-proton cycle

In stars with hotter cores (T > 15 x 106 K), the carbon-cycle dominates

Proton-Proton Cycle The proton-proton cycle is a

series of three nuclear reactions believed to operate in the Sun

The net result is the joining of four protons to form a helium-4 nucleus

Energy liberated is primarily in the form of em radiation (gamma rays), positrons and neutrinos

HHHeHeHe

or

eHeHeH

Then

HeHH

eHHH

11

11

42

32

32

42

32

11

32

21

11

21

11

11

The Carbon Cycle Hydrogen nuclei can

fuse into nuclei heavier than helium

Multiple steps lead to the release of energy

The net effect is that four hydrogen nuclei combine to form a helium-4 nucleus

The Carbon-12 is returned at the end, it acts as a catalyst

1 12 131 6 7

13 127 6

1 13 141 6 7

1 14 151 7 8

15 158 7

1 15 12 41 7 6 2

H C N

N C e

H C N

H N O

O N e

H N C He

Energy in a Star Depending on its mass, a star transforms energy at the

rate of 1023 to 1033 W The energy from the core is transferred outward through

the surrounding layers Neutrinos carry energy directly through these layers to space Energy carried by photons is absorbed by the gasses in the

layer outside the core and gradually works it way to the surface by convection

This energy is emitted from the surface in the form of em radiation, mainly infrared, visible and ultraviolet

The star is stable as long as its supply of hydrogen in the core lasts