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Nuclear Binding, Radioactivity

SPH4UI Physics Modern understanding: the ``onion’’ picture

Let’s see what’s inside!

AtomNucleusProtons and neutrons

3

Nice Try

Introduction: 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)

1911 Rutherford, Geiger and Marsden performed scattering experiments Established the point mass nature of the nucleus

Nuclear force was a new type of force

1919 Rutherford and coworkers first observed nuclear reactions in which naturally occurring alpha particles bombarded nitrogen nuclei to produce oxygen

1932 Cockcroft and Walton first used artificially accelerated protons to produce nuclear reactions

1932 Chadwick discovered the neutron

1933 the Curies discovered artificial radioactivity

1938 Hahn and Strassman discovered nuclear fission

1942 Fermi achieved the first controlled nuclear fission reactor

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Some Properties of Nuclei All nuclei are composed of protons and neutrons

Exception is ordinary hydrogen with just a proton

The atomic number (charge), Z, equals the number of protons in the nucleus

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

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

Notation

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

A

Z X where X is the chemical symbol of the element

27

13 Al

Charge and mass

Charge: The electron has a single negative charge, -e (e = 1.60217733 x 10-19 C)

The proton has a single positive charge, +e

Thus, charge of a nucleus is equal to Ze

The neutron has no charge

Makes it difficult to detect

Mass: It is convenient to use atomic mass units, u, to express masses

1 u = 1.660559 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

Summary of Masses

Masses

Particle kg u MeV/c2

Proton 1.6726 x 10-27 1.007276 938.28

Neutron 1.6750 x 10-27 1.008665 939.57

Electron 9.101 x 10-31 5.486x10-4 0.511

What is the order of magnitude of the number of protons in your body?

Of the number of neutrons? Of the number of electrons? Take your

mass approximately equal to 70 kg.

Quick problem: protons in your

body

An iron nucleus (in hemoglobin) has a few more neutrons

than protons, but in a typical water molecule there are

eight neutrons and ten protons. So protons and neutrons

are nearly equally numerous in your body, each

contributing 35 kg out of a total body mass of 70 kg.

Same amount of neutrons and electrons.

28

27

1nucleon35 10 protons

1.67 10N kg

kg

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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

The KE of the particle must be completely converted to PE

2

2

4 ek Zed

mv 2 1 2

21

2e e

e Zeq qmv k k

r d or

For gold: d = 3.2 x 10-14 m, for silver: d = 2 x 10-14 mSuch small lengths are often expressed in femtometers where 1 fm = 10-15 m

(also called a fermi)

Size of Nucleus

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

13

or r A

27

13 Al has radius 153.6 10r m

A

Z

Example:

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 (or strong) 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

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Nuclear Stability chart

Light nuclei are most stable if N = Z

Heavy nuclei are most stable when N > Z

As the number of protons increase, the Coulomb force increases and so more nucleons are needed to keep the nucleus stable

No nuclei are stable when Z > 83

Isotopes

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

Example: 11

6C14

6C13

6C12

6C

A

Z Xatomic number (charge), Zneutron number, Nnucleon number, A,

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 separated protons

and neutrons

Binding Energy per Nucleon

Iron (Fe) is most binding energy/nucleon. Lighter have too

few nucleons, heavier have too many.

BIN

DIN

G E

NE

RG

Y in

Me

V/n

ucle

on

Binding Energy Plot

Fission

Fusion = Combining small atoms into large

Fission = Breaking large atoms into small

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Binding Energy Notes

Except for light nuclei, the binding energy is about 8 MeV per nucleon

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 curve is slowly varying at A > 40

This suggests that the nuclear force saturates

A particular nucleon can interact with only a limited number of other nucleons

Question

Where does the energy released in the nuclear

reactions of the sun come from?

(1) covalent bonds between atoms

(2) binding energy of electrons to the nucleus

(3) binding energy of nucleons

Which element has the highest binding energy/nucleon?

Question

• Neon (Z=10)

• Iron (Z=26)

• Iodine (Z=53)

Which of the following is most correct for the total

binding energy of an Iron atom (Z=26)?

9 MeV

234 MeV

270 MeV

504 Mev

For Fe, B.E./nucleon 9MeV

56

26 Fe has 56 nucleons

Total B.E 56x9=504 MeV

Question

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Binding Energy

Einstein’s famous equation E = m c2

Proton: mc2 = 938.3MeV

Neutron: mc2= 939.5MeV

Deuteron: mc2 =1875.6MeV

Adding these, get

1877.8MeV

Difference is

Binding energy,

2.2MeV

MDeuteron = MProton + MNeutron – |Binding Energy|

Big Problem: binding energy

Calculate the average binding energy per nucleon of 93

41 Nb

Using atomic mass units

Calculate the average binding energy per nucleon of

Given:

mp= 1.007276u

mn= 1.008665u

Find:

Eb = ?

In order to compute binding energy, let’s first find the

mass difference between the total mass of all protons and

neutrons in Nb and subtract mass of the Nb:

41pN

Thus, binding energy is

2 0.842519 931.58.44 nucleon

93b

m c u MeV uE MeV

A

93

41 Nb

93 41 52nN

Number of protons:

Number of neutrons:

Mass difference:

41 52p n Nbm m m m

41 1.007276 52 1.008665 92.9063768

0.8425192

u u u

u

1u=931.5 MeV

Radioactivity

Radioactivity is the spontaneous emission of radiation

Experiments suggested that radioactivity was the result of the

decay, or disintegration, of unstable nuclei

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

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a particles: nuclei4

2 He

b particles: electrons

g photons (more energetic than x-rays)

Types of Radioactivity

Barely penetrate a piece of paper

Can penetrate a few mm of aluminum

Radioactive

sources

B field into

screen

detector

Can penetrate

several cm of

lead

The Decay Processes –

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

Conservation of mass-energy and conservation of momentum

must hold

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 4

2 2

A A

Z ZX Y He

Alpha Decay -- Example

Decay of 226Ra

Half life for this decay is 1600 years

Excess mass is converted into kinetic energy

Momentum of the two particles is equal and opposite

226 222 4

88 86 2Ra Rn He

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Beta Decay

During beta decay, the daughter nucleus has the same number of

nucleons as the parent, but the atomic number is one less

In addition, an electron (positron) was observed

The emission of the electron is from the nucleus

The nucleus contains protons and neutrons

The process occurs when a neutron is transformed into a proton

and an electron

Energy must be conserved

Beta Decay – Electron Energy The energy released in the decay

process should almost all go to kinetic energy of the electron

Experiments showed that fewelectrons had this amount of kinetic energy

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

Beta Decay

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

A A

Z Z

A A

Z Z

X Y e

X Y e

Gamma Decay

Gamma rays are given off when an excited nucleus “falls” to a lower energy state

Similar to the process of electron “jumps” to lower energy states and giving off photons

The excited nuclear states result from “jumps” made by a proton or neutron

The excited nuclear states may be the result of violent collision or more likely of an alpha or beta emission

Example of a decay sequence

The first decay is a beta emission

The second step is a gamma emission

The C* indicates the Carbon nucleus is in an excited state

Gamma emission doesn’t change either A or Z

12 12

5 6

12 12

6 6

*

*

B C e

C C

g

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238 234

92 90U Th a a: example 4

2 He arecall

b: example

Decay Rules

1) Nucleon Number is conserved.

2) Atomic Number (charge) is conserved.

3) Energy and momentum are conserved.

g: example * 0

0

A A

Z ZP P g

1) 238 = 234 + 4 Nucleon number conserved

2) 92 = 90 + 2 Charge conserved

1 1 0

0 1 1n p e

Neutrino needed to

conserve energy and

momentum.

0

0

A nucleus undergoes a decay. Which of the following is FALSE?

1. Nucleon number decreases by 4

2. Neutron number decreases by 2

3. Charge on nucleus increases by 2

Practice

a decay is the emission of4

2 He a

238 234 4

92 90 2U Th He Ex.

Z decreases by 2

(charge decreases!)

A decreases by 4

b The nucleus undergoes decay.

234

90Th

Which of the following is true?

1. The number of protons in the daughter

nucleus increases by one.

2. The number of neutrons in the daughter

nucleus increases by one.

b decay is accompanied by the emission of an

electron: creation of a charge -e.

In fact, inside the nucleus, and the electron and

neutrino “escape.”en p e

Practice

000

1

???

??

234

90 XTh

e 000

1

234

91

234

90 PaTh

e

Decay

Which of the following decays is NOT allowed?

214 210 4

84 82 2Po Pb He

238 234

92 90U Th a

40 40 0 0

19 20 1 0K p e

14 14

6 7C N g

1

2

3

4

238 = 234 + 4

92 = 90 + 2

214 = 210 + 4

84 = 82 + 2

14 = 14+0

6 <> 7+0

40 = 40+0+0

19 = 20-1+0

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Which of the following are possible reactions?

(a) and (b). Reactions (a) and (b) both conserve total charge and total

mass number as required. Reaction (c) violates conservation of mass

number with the sum of the mass numbers being 240 before reaction

and being only 223 after reaction.

Decay 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

The decay rate or activity, R, of a sample is defined as the number

of decays per second

NR N

t

N N t

Radioactivity

NN

t

No. of nuclei

present

decay constant

Decays per second,

or “activity”

Start with 16 14C atoms.

After 6000 years, there are only 8 left.

How many will be left after another 6000 years?

1) 0 2) 4 3) 8

Every 6000 years ½ of atoms decay

Decay Curve

The decay curve follows the

equation

The half-life is also a useful

parameter

The half-life is defined as the

time it takes for half of any

given number of radioactive

nuclei to decay

693.02lnT 21

0

tN N e

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Units

The unit of activity, R, is the Curie, Ci

1 Ci = 3.7 x 1010 decays/second

The SI unit of activity is the Becquerel, Bq

1 Bq = 1 decay / second Therefore, 1 Ci = 3.7 x 1010 Bq

The most commonly used units of activity are the mCi and the µCi

time

1/2

0 0

1( )

2

t

TtN t N e N

Decay Function

Practice

The half-life for beta-decay of 14C is ~6,000 years. You

test a fossil and find that only 25% of its 14C is un-decayed.

How old is the fossil?

3,000 years

6,000 years

12,000 years

At 0 years: 100% remains

At 6,000 years: 50% remains

At 12,000 years: 25% remains

Instead of base e we can use base 1/2:

0( ) tN t N e Survival:

No. of nuclei present

at time t

No. we started

with at t=0

1/21

2

t

Tte

1/2

0.693T

where

Then we can write1/2

0 0

1( )

2

t

TtN t N e N

Half life

Radioactivity Quantitatively

NN

t

No. of nuclei

present

decay constant

Decays per second,

or “activity”

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Radioactivity Example

1/2

0 0

1( )

2

t

TtN t N e N

1/2

0.693T

The half-life for beta-decay of 14C is 5730 years. If you start

with 1000 carbon-14 nuclei, how many will be around in

22920 years?

1/2

22920

573

0

011000

2

.5

)

62

1(

2

t

T

N t N

4

1/2

0.693

5730

1.209 10

0.693T

41.209 10 22920)

0

1000

62.5

tN t N e

e

Uses of Radioactivity

Carbon Dating

Beta decay of 14C is used to date organic samples

The ratio of 14C to 12C is used

Smoke detectors

Ionization type smoke detectors use a radioactive source to ionize the air in a chamber

A voltage and current are maintained

When smoke enters the chamber, the current is decreased and the alarm sounds

Radon pollution

Radon is an inert, gaseous element associated with the decay of radium

It is present in uranium mines and in certain types of rocks, bricks, etc that may be used in home building

May also come from the ground itself

Binding Energy

Which system “weighs” more?

1) Two balls attached by a relaxed spring.

2) Two balls attached by a stretched spring.

3) They have the same weight.

M1 = Mballs + Mspring

M2 = Mballs + Mspring + Espring/c2

M2 – M1 = Espring/c2

≈ 10-16 Kg

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Strong Nuclear Force

Acts on Protons and Neutrons

Strong enough to overcome Coulomb

repulsion

Acts over very short distances

Two atoms don’t feel force

Q Values (nice to know)

Energy must also be conserved in nuclear reactions

The energy required to balance a nuclear reaction is called the Q value of the 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

Problem: nuclear reactions

Determine the product of the reaction

What is the Q value of the reaction?7 4

3 2 ?Li He n

Determine the product of the reaction

What is the Q value of the reaction?

Given:

reaction

Find:

Q = ?

In order to balance the reaction, the total amount

of nucleons (sum of A-numbers) must be the

same on both sides. Same for the Z-number.

7 4 1 10X X

The Q-value is then

7 4 10

2 2

7.016005 4.002603 10.012938 1.008665 931.5 /

2.79

nLi He BQ m c m m m m c

u u u u Mev u

MeV

3 2 0 5Y Y

Number of nucleons (A):

Number of protons (Z):

Thus, it is B, i.e.7 4 10 1

3 2 5 0Li He B n

7 4 1

3 2 0?X

YLi He n

It is easier to

use atomic

mass units

rather than kg.

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Summary

Nuclear Reactions Nucleon number conserved

Charge conserved

Energy/Momentum conserved

a particles = nucleii

b- particles = electrons

g particles = high-energy photons

Decays Half-Life is time for ½ of atoms to decay

0( ) tN t N e Survival: 1/2

0.693T

4

2 He