Chapter 26 Particle and Nuclear Physics (A2)

CAMBRIDGE A – LEVEL

PHYSICS

PARTICLE AND

NUCLEAR PHYSICS

(A2)

L E A R N I N G O U TC O M E S

No.

LEARNING OUTCOME

i

_{A p p l y t h e m a s s – e n e r g y c o n s e r v a t i o n p r i n c i p l e t o}

c a l c u l a t e t h e r e a c t i o n e n e r g y i n a n u c l e a r p r o c e s s .

_{U n d e r s t a n d w h a t i s m e a n t b y m a s s e x c e s s a n d t h e}

r e l a t i o n s h i p a n d m a s s e x c e s s a n d n u c l e a r

p r o c e s s e s .

_{R e l a t e t h e c o n c e p t o f b i n d i n g e n e r g y w i t h t h e}

m a s s d e f e c t .

_{R e l a t e b i n d i n g e n e r g y a n d b i n d i n g e n e r g y p e r}

n u c l e o n .

_{G r a p h t h e r e l a t i o n s h i p b e t w e e n b i n d i n g e n e r g y}

p e r n u c l e o n a g a i n s t n u c l e o n n u m b e r.

L E A R N I N G O U T C O M E S

No.

_{LEARNING OUTCOME}

vi

D i f f e r e n t i a t e b e t w e e n n u c l e a r f i s s i o n a n d f u s i o n .

vii

U n d e r s t a n d t h e t e r m a c t i v i t y a n d d e c a y c o n s t a n t .

viii

U s e t h e e x p o n e n t i a l d e c a y m e t h o d t o c a l c u l a t e t h e

n u m b e r o f u n d e c a y e d n u c l e i a n d a c t i v i t y.

ix

U n d e r s t a n d w h a t i s m e a n t b y h a l f l i f e a n d

c a l c u l a t e t h e h a l f l i f e o f a r a d i o a c t i v e s a m p l e .

U N I T S O F M A S S

• As we have learned at AS Level,

the SI unit for mass is the

kilogram.

• Physicists have also developed an

alternative unit for mass, the

U N I T S O F M A S S



• 1 u

is equivalent to

_

the

mass

of

1

atom

of

the





_isotope,

_{in kilograms}

_.

• The mass of one atom of



_

=

U N I T S O F M A S S



• Hence,



kg

.

• The table on the next slide shows

the masses of some light nuclides

in atomic mass units.

Table 43.2, page 1441, Section 43.1: Properties of Nuclei; Chapter 43: Nuclear Physics; Sear’s and Zemansky’s University Physics, Young and Freedman, 13th edition, Pearson Education, San Francisco, 2012.

U N I T S O F M A S S

U N I T S O F M A S S

• The table below lists the masses

of the nucleons and the electron

in atomic mass units:

Particle

Mass (u)

Proton

1.007276

Neutron

1.008665

Electron

0.000549

M A S S - E N E R G Y

C O N S E R VAT I O N

M A S S - E N E R G Y

C O N S E R VAT I O N

• As we have learned at AS

Level,

mass

–

energy

is

conserved

in

a

nuclear

process

, but, however,

mass

itself is not conserved

.

M A S S - E N E R G Y

C O N S E R VAT I O N

M A S S - E N E R G Y

C O N S E R VAT I O N

• The

difference between the total

mass after the reaction and the

total mass before the reaction

gives us an idea of the amount of

M A S S - E N E R G Y

C O N S E R VAT I O N

M A S S - E N E R G Y

C O N S E R VAT I O N



.  





 



.

• We may use Einstein’s mass – energy

equation



; where:

  reaction energy

, in J;

  ∑ 



 ∑ 



, in

kg, and

 speed of light, .  





 



.

M A S S - E N E R G Y

C O N S E R VAT I O N

M A S S - E N E R G Y

C O N S E R VAT I O N

• What if we have masses in atomic

• What if we have masses in atomic

mass units (u)?

• We now use

,

where:

E = energy, MeV

, and

m = mass, in u

.

931.494 MeV/u is a conversion

M A S S - E N E R G Y

C O N S E R VAT I O N

M A S S - E N E R G Y

C O N S E R VAT I O N

• We can also use a different

version of Einstein’s equation:



• This version

relates the change in

EX A M P L E

Table 43.1, page 1441, Section 43.1: Properties of Nuclei; Chapter 43: Nuclear Physics; Sear’s and Zemansky’s University Physics, Young and Freedman, 13th

M A S S E XC ES S

mass

mass

mass

mass excess

excess

excess

excess 

 mass

mass

mass

mass #in

#in u'

#in

#in

u'

u'

u' 





 neutron

neutron

neutron number

neutron

number

number

number

• The

mass excess of a nuclide is the

difference between its actual mass,

in atomic mass units (u) and its

mass number

.

• Mathematically:

mass

mass

mass

M A S S E XC ES S

Figure 31.3, page 493, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

M A S S E XC ES S

• The table on the previous slide gives

some nuclides and their masses.

Question 7, page 494, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

M A S S E XC ES S

We can use mass excess values to find

• We can use mass excess values to find

out whether a nuclear reaction is

feasible.

• We can do this by calculating:

I.

the total mass excess of the reactants,

II.

the total mass excess of the products,

and

M A S S E XC ES S

• If the

total mass excess of the

products is smaller than the total

mass excess of the reactants

, the

reaction is feasible

.

• On the other hand,

if the converse is

true, then the reaction will not be

feasible

.

M A S S E XC ES S

,

-

_→

1

_/0

₂

5

₃₄

_{2 37}

• Let us have a look at an example:

,

-

_→

1

_/0

₂

5

₃₄

_{2 37}

Nuclide

Mass Excess (u)

,

-

_+0.045563

/0

1

_−0.073843

34

5

_−0.085588

7

+0.008665

M A S S E XC ES S

 0.073843 2 0.085588 2 3 

0.008665  0.133436 u

• Using the values from the table in the

previous slide, we can obtain:

I. Total

mass

excess

of

reactants

u

II. Total

mass

excess

of

products

 0.073843 2 0.085588 2 3 

M A S S E XC ES S

• In this example, the total mass

excess of the reactants is larger than

the

total

mass

excess

of

the

products.

• This means that

the products will be

more stable than the reactants

.

Hence, this

reaction will occur

.

M A S S D E F EC T



• If we were to measure the mass

of

one

nucleus

of

the



_

isotope, it will be different from

the total mass of the 6 neutrons

and the 6 protons make up the

nucleus.

Figure 31.1, page 492, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

Cambridge University Press, Cambridge, UK,2014.

M A S S D E F EC T

• By using the values from the

table on the previous slide, we

will obtain a mass defect of



_{kg for this}

carbon nuclide.

M A S S D E F EC T

• Definition: “

The mass defect is

the difference between the

total mass of the individual

nucleons and the mass of the

nucleus

.”

M A S S D E F EC T

• Why is there a mass defect?

• Why is there a mass defect?

• This mass defect exists because

when the individual nucleons

formed the nucleus, some of the

mass

was

converted

into

potential energy that is used to

hold the nucleons together.

B I N D I N G E N E R GY

• The strong force is responsible in

• The strong force is responsible in

producing

this

change

in

potential energy and thus binds

the nucleons.

• Work must be done to separate

these

nucleons

apart

to

an

infinite amount of separation.

B I N D I N G E N E R GY

• Definition:

“

The

binding

energy of a nucleus is the

energy required to separate

all the nucleons in a nucleus

to

an

infinite

amount

of

separation

.”

B I N D I N G E N E R GY

• By using mass – energy equivalence,

we can use the mass defect and

convert it into the binding energy.

• To achieve this, use this conversion

factor:

of mass defect is

equivalent to

of

EX A M P L E

Exercise 43.39, page 1476; Chapter 43: Nuclear Physics; Sear’s and Zemansky’s University Physics, Young and Freedman, 13th _{edition, Pearson Education, San}

B I N D I N G E N E R GY

Examples from Page 367; Section 13.7: The Mass Defect, Chapter 13: Nuclear Physics;

International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United Kingdom, 2008.

B I N D I N G E N E R G Y P E R

N U C L E O N

B I N D I N G E N E R G Y P E R

N U C L E O N

• The

binding energy per nucleon

is equal to the binding energy of

the nucleus divided by the total

number of nucleons present.

B I N D I N G E N E R G Y P E R

N U C L E O N

B I N D I N G E N E R G Y P E R

N U C L E O N

• Why is the binding energy per

• Why is the binding energy per

nucleon important?

• The binding energy per nucleon

value gives us the stability of that

nuclide relative to its neighbours,

i.e. how hard is it for that nuclei to

radioactively

decay

into

its

neighbours.

B I N D I N G E N E R G Y P E R

N U C L E O N

B I N D I N G E N E R G Y P E R

N U C L E O N

B I N D I N G E N E R G Y P E R

N U C L E O N

B I N D I N G E N E R G Y P E R

N U C L E O N

The graph on the previous slide shows a

• The graph on the previous slide shows a

sketch of the binding energy per nucleon

for several elements.

• The

higher the value of the binding

energy per nucleon compared to its

neighbours, the harder it is for that

nuclei to decay radioactively into one of

its neighbours.

B I N D I N G E N E R G Y P E R

N U C L E O N

B I N D I N G E N E R G Y P E R

N U C L E O N

?

• The nuclide with the

highest

binding energy per nucleon is



@

_.

• Peaks representing

_

5

,



_

and

?



_{indicate nuclides that are}

EX A M P L E

Questions 9 amd 10, page 496, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

NU C L EA R FI S S I O N

• Nuclear fission is a decay

process in which an unstable,

heavy nucleus splits into two

fragments of almost the same

mass

.

NU C L EA R FI S S I O N

• The two fragments, known as

fission

fragments will have a higher binding

energy per nucleon as compared to

the parent nucleus

.

• Nuclear

fission

is

achieved

by

bombarding the heavy nucleus with

a neutron

.

NU C L EA R FI S S I O N

• An example of a fission reaction

is seen below:

Page 370; Section 13.10: Nuclear Fission, Chapter 13: Nuclear Physics; International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United Kingdom, 2008.

NU C L EA R FI S S I O N

• Fission

reactions

are

accompanied by the release of

energy

because

the

binding

energy per nucleon after the

reaction is higher than that

NU C L EA R FU S I O N

• Nuclear fusion occurs when

two or more small nuclei

come together (fuse) to form

a larger nucleus.

NU C L EA R FU S I O N

• Nuclear fusion only

occurs under

conditions of high pressure and

temperature;

e.g. on the Sun’s

NU C L EA R FU S I O N

The examples below are of nuclear fusion

• The examples below are of nuclear fusion

reactions:

Page 1469, Section 43.8: Nuclear Fussion; Chapter 43: Nuclear Physics; Sear’s and Zemansky’s University Physics, Young and Freedman, 13th _{edition, Pearson}

F U S I O N v s . F I S S I O N

Figure 31.6, page 496, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

F U S I O N v s . F I S S I O N

• As seen on the graph on the previous

• As seen on the graph on the previous

slide, fusion and fission occur in

order to increase the binding energy

per nucleon.

• Nuclei

in between A and B tend to

undergo fusion

,

while

nuclei in

between B and C tend to undergo

fission

.

E X A M P L E S

Question 16; Set 45: Structure of the Nucleus and Radioactivity; page 228; PROBLEMS IN PHYSICS ; E.D GARDINER, B.L McKITTRICK; McGraw – Hill Book Company, Sydney 1985.

E X A M P L E S

Question 17; Set 45: Structure of the Nucleus and Radioactivity; page 228; PROBLEMS IN PHYSICS ; E.D GARDINER, B.L McKITTRICK; McGraw – Hill Book Company, Sydney 1985.

E X A M P L E S

R A D I AT I O N D E T E C T I O N

• Outlined below are methods of radiation

detection:

R A D I AT I O N D E T E C T I O N

Examples from Pages 355 - 357; Section 13.14: Detecting Radioactivity; Chapter 13: Nuclear Physics; International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United Kingdom, 2008.

N U C L EA R D EC AY

• If we were to use a GM

• If we were to use a GM

counter

to

measure

radioactivity by listening to the

number of clicks, we will not

be able to predict when the

next click is heard.

N U C L EA R D EC AY

• If we were to use a ratemeter,

• If we were to use a ratemeter,

the reading on the ratemeter

fluctuates up and down.

• This occurs because of the

random

and

spontaneous

nature of nuclear processes

.

N U C L EA R D EC AY

• It is

random because

• It is

random because

i.

we

cannot predict which

nucleus in a sample will

decay

, and

ii. The

probability that each of

the nuclei will decay in per

unit of time

is

constant

. This

probability is known as the

decay constant, λ.

N U C L EA R D EC AY

• If we plot a graph of count

• If we plot a graph of count

rate vs. time, we would

obtain a graph as seen on

the next slide.

• The fluctuations indicate

the

random

nature

of

N U C L EA R D EC AY

Figure 31.8, page 497, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

Cambridge University Press, Cambridge, UK,2014.

N U C L EA R D EC AY

• Nuclear decay is

spontaneous

• Nuclear decay is

spontaneous

because

:

i.

its

occurrence

is

independent of any external

or

environmental

factors,

and

ii. not affected by the presence

ACTIVITY

ACTIVITY

• Radioactive

nuclei

undergo

decay.

• Therefore,

the

amount

of

parent nuclei reduces with

time.

ACTIVITY

ACTIVITY

AB

AC

AB

AC

• The

rate of nuclei decay,

AB

_AC

is

directly proportional to the

amount of undecayed nuclei

present in the sample

,

.

• We can rewrite this expression

AB

• We can rewrite this expression

as

AB

AC

where:

decay constant

that has

units of







.

the minus sign indicated this is

a decay.

ACTIVITY

ACTIVITY

• Definition

: “

The decay constant,

AC

• Definition

: “

The decay constant,

is defined as the probability

per unit time interval that the

nuclei will undergo decay

.”

•

AB

_AC

is also known as the

activity of the source,

.

ACTIVITY

ACTIVITY

• Definition

:

“

The

activity

of

a

radioactive source is the number of

nuclear decays produced per unit of

time in the source”

.

• Activity is

measured in Becquerels

(Bq)

, and

1 Becquerel is 1 decay per

second

.

ACTIVITY

ACTIVITY

• By

combining

the

equations

AB

AB

D 

E



.

• By

combining

the

equations

AB

AC

and

AB

AC

, we will

obtain

, where:

 

activity of the sample, in Bq;

B 

number of undecayed nuclei,

and

D 

decay constant, in

E



.

ACTIVITY

ACTIVITY

E X A M P L E S

Questions 12 and 13, page 499, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

E X A M P L E S

Questions 15 and 16, page 501, Chapter 31: Nuclear Physics; Cambridge

International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

Cambridge University Press, Cambridge, UK,2014.

• The

solution

of

the

equation

is in

the form of

.

HALF LIFE

HALF LIFE

• The

quantities

number

of

undecayed nuclei,

activity,

and received count rate,

all

have

the

general

form



DC

.

HALF LIFE

HALF LIFE

• We now have three equations





DC

• We now have three equations

that

relate

these

three

quantities with time,

DC

_

DC

_

DC

HALF LIFE

HALF LIFE

B  B

_

F

DC

λ

• If we plot the equation

B  B



F

DC

,

for

three different values of

λ ,we would

obtain:

HALF LIFE

HALF LIFE

Cambridge University Press, Cambridge, UK,2014.



H

I

_J

• Definition

: “

The half life,



H

, of a

radioactive nuclide is the time taken

for the number of undecayed nuclei

to be reduced to half its original

number

”.

• How do we calculate the value of

I

K

J

⁄

?

HALF LIFE

HALF LIFE

K

J

⁄



_

M



H

_D

• At

K

J

⁄

,



_

M

. When we

substitute

into

the

equation



DC

, we obtain



_

NO

K

H

J

.

• By taking the natural logarithms on

both sides, we get



H

.P

D

HALF LIFE

HALF LIFE

E X A M P L E S

Question 18, page 501, Chapter 31: Nuclear Physics; Cambridge International AS and A Level Physics Coursebook, Sang, Jones, Chadha and Woodside, 2nd _edition,

Cambridge University Press, Cambridge, UK,2014.

E X A M P L E S

E X A M P L E S

H O M E WO R K

1. Question 8, Paper 4, Summer 2008.

1. Question 8, Paper 4, Summer 2008.

2. Question 9, Paper 4, Summer 2009.

3. Question 8, Paper 41, Winter 2009.

4. Question 8, Paper 42, Winter 2009.

5. Question 8, Paper 41, Summer 2010.

6. Question 8, Paper 42, Summer 2010.

7. Question 8, Paper 41, Winter 2010.

8. Question 8, Paper 41, Summer 2011.

H O M E WO R K

9. Question 8, Paper 42, Summer 2011.

10.Question 8, Paper 41, Winter 2011.

11.Question 8, Paper 43, Winter 2011.

12.Question 9, Paper 41, Summer 2012.

13.Question 8, Paper 42, Summer 2012.

14.Question 8, Paper 43, Winter 2012.