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

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 27Atomic number is 13Contains 13 protonsContains 14 (27 – 13) neutrons

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

AZ X where X is the chemical symbol of the element

2713 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.

2827

1 nucleon35 10 protons

1.67 10N kg

kg

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

2 e 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 m

Such 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

2713 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

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: 116C

146C

136C

126C

AZ X

atomic 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 M

eV

/nucl

eon

Binding Energy Plot

Fission

Fusi

on Fusion = Combining small atoms into large

Fission = Breaking large atoms into small

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

5626Fe has 56 nucleons

Total B.E 56x9=504 MeV

Question

Binding Energy

Einstein’s famous equation E = m c2

Proton: mc2 = 938.3MeVNeutron: 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 9341Nb

Using atomic mass units

Calculate the average binding energy per nucleon of

Given:

mp= 1.007276umn= 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

9341Nb

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 particlesThe particles are 4He nuclei

Beta particlesThe 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 raysThe “rays” are high energy photons

a particles: nuclei42He

b- particles: electrons

g photons (more energetic than x-rays)

Types of Radioactivity

Barely penetrate a piece of paperCan 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 42 2

A AZ 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

226 222 488 86 2Ra Rn He

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 few electrons 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 AZ Z

A AZ 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

12 125 6

12 126 6

*

*

B C e

C C

238 23492 90U Th : example 4

2He recall

: example

Decay Rules

1) Nucleon Number is conserved.2) Atomic Number (charge) is conserved.3) Energy and momentum are conserved.

g: example * 00

A AZ ZP P

1) 238 = 234 + 4 Nucleon number conserved

2) 92 = 90 + 2 Charge conserved1 1 00 1 1n p e

Neutrino needed to conserve energy and momentum.

00

A nucleus undergoes 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

decay is the emission of42He

238 234 492 90 2U Th He Ex.

Z decreases by 2(charge decreases!)

A decreases by 4

The nucleus undergoes decay. 234

90ThWhich 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.

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

0001

?????

23490 XTh

e 0001

23491

23490 PaTh

e

Decay

Which of the following decays is NOT allowed?

214 210 484 82 2Po Pb He

238 23492 90U Th

40 40 0 019 20 1 0K p e

14 146 7C N

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+019 = 20-1+0

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) 8Every 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

0tN N e

Units

The unit of activity, R, is the Curie, Ci1 Ci = 3.7 x 1010 decays/second

The SI unit of activity is the Becquerel, Bq1 Bq = 1 decay / second

Therefore, 1 Ci = 3.7 x 1010 BqThe 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 QuantitativelyN

Nt

No. of nuclei present

decay constant

Decays per second, or “activity”

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

TN 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

Strong Nuclear Force

Acts on Protons and Neutrons

Strong enough to overcome Coulomb repulsion

Acts over very short distancesTwo 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 reactionThere is a mass “loss” in the reactionThere is a release of energyQ is positive

An endothermic reactionThere is a “gain” of mass in the reactionEnergy is needed, in the form of kinetic energy of the incoming

particlesQ is negative

Problem: nuclear reactions

Determine the product of the reaction What is the Q value of the reaction?

7 43 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 13 2 5 0Li He B n

7 4 13 2 0?X

YLi He n

It is easier to use atomic mass units rather than kg.

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

42He