NE 301 - Introduction to Nuclear Science

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NE
301
-

Introduction to Nuclear
Science

Spring
2012

Classroom Session
5:




Isotopes and Decay Diagrams


Nuclear Reactions


Energy of nuclear reactions


Neutron Cross Sections


Activation Calculations


Radioactive Decay and Growth


Reminder

Load
TurningPoint


Reset slides


Load List


2

Page 98
-
Shultis


-
,

+

produce three products:


Cannot

say
energy

of


Neutrinos by Fermi (1933)

We
only

can say
maximum

energy of


Binary Nuclear Reactions

Binary = 2 reactants
(many times 2 products too)

Most important type of nuclear reaction

Most elements produced by binary
rxns
.
in stars


Nomenclature:



4

x X Y y
  
Light nuclide usually
projectile

Heavy nuclide
usually
target

Heavy
Product

Light
Product

5

For Binary Reactions:
x +X


夠⬠Y

x is a projectile with KE (E
x
).
X

is a target
stationary

nucleus E
X
=0





simplification

y



Cos
y

0, Real
y
E

6

y



Cos
y

0, Real
y
E

A 5.5
MeV



particle is incident on Li causing
7
Li(

,n
)
10
B.
What is the KE of neutron scattered 30
o
?

A 5.5
MeV



particle is incident on Li causing
7
Li(

,n
)
10
B.
What is the KE of neutron scattered 30
o
?

7

0, Real
y
E

y



Cos
y

1.
0
MeV

2.
0.31
MeV

3.
1.31
MeV

4.
2.31
MeV

5.
3.31
MeV

6.
5.5
MeV

7
Li(alpha,n)
10
B

8

FIRST BALANCE THE EQUATION!!!

Endothermic
Rxn

Neutron Energy = 1.31MeV

What would be the neutron energy if incident alpha
particle is
1MeV

instead?

Can’t happen…

Solution exists
only

if

0, Real
y
E

Potential “

” Factors


Q<0


Heavy projectiles (
m
Y
-
m
x
<0
)


Large scattering angles Cos

<0

Big enough E
x

can guarantee


Physical meaning:
Threshold Energy

0, Real
y
E

Argument of
root >0

10

Kinematic Threshold (only if Q<0)

Arises from conservation of:


Energy


Linear momentum

(Details are in the textbook)

In most nuclear reactions (m
i
’s>Q), the
kinematic threshold simplifies to:






ONLY FOR Q<0
i.e. endothermic
rxns
.


E
x
th


1

m
x
m
X






Q
What is the kinematic threshold for:
7
Li(

,n
)
10
B ?

11

0.4 MeV
1.4 MeV
2.4 MeV
3.4 MeV
4.4 MeV
17%
0%
83%
0%
0%
1.
0.4
MeV

2.
1.4
MeV

3.
2.4
MeV

4.
3.4
MeV

5.
4.4
MeV


E
x
th


1

m
x
m
X






Q
12

7 10
Li (,n) B
(4.002603 7.016004-1.008665-10.012937) 931
.494 -2.790 MeV
4.002603
1 - 1 (-2.790) 4.382 MeV
7.016004
th
x
x
X
Q
m
E Q
m

   
 
 
     
 
 
 
 
What is the kinematic threshold for:
13
C(
d,t
)
12
C

13

-1.5 MeV
0 MeV
1.5 MeV
3 MeV
9.6 MeV
0%
0%
0%
0%
0%

E
x
th


1

m
x
m
X






Q
1.
-
1.5
MeV

2.
0
MeV

3.
1.5
MeV

4.
3
MeV

5.
9.6
MeV

Remember: Kinematic Threshold
only

for
Endothermic

Reactions

Balance. Then:

(13.003355+2.014102
-
12
-
3.016049)*931.494

Q=1.31154
MeV

14

Remember: Kinematic Threshold
only

for
Endothermic

Reactions

Exothermic = Kinematic Threshold is 0
MeV

15

Coulomb

Barrier Threshold (fig. 3.9)

ONLY

when the incident
nuclide is
charged

All nuclides are positive


Projectile needs energy
to overcome electrostatic
repulsion


1/3 1/3
1.2 [MeV]
C
x X
x
x X
Z Z
E
A A


Engineering Equation.
MeV

units already worked out (don’t worry)




NOT for
incident

NEUTRONS nor

’s

Binding
Energy

Coulombic

Threshold

What is the coulomb barrier
threshold for:
7
Li(

,n
)
10
B ?

16

1 MeV
2 MeV
3 MeV
4 MeV
5 MeV
0%
0%
0%
0%
0%
1.
1
MeV

2.
2
MeV

3.
3
MeV

4.
4
MeV

5.
5
MeV

1/3 1/3
1.2 [MeV]
C
x X
x
x X
Z Z
E
A A


Clicker Answer

17

1.2
3
2
7
3
4
3
2.057
MeV
What is the coulomb barrier
threshold for:
14
N(
n,

)
11
B ?

18

0 MeV
1 MeV
2 MeV
3 MeV
Q MeV
0%
0%
0%
0%
0%
1.
0
MeV

2.
1
MeV

3.
2
MeV

4.
3
MeV

5.
0.98 x Q MeV

19

Overall Threshold Energy

Neutral Incident particle=No Coulomb
Barrier.

Q>0 = No Kinetic Threshold

Charged particles and Q<0 = both
thresholds apply, and:







Do
NOT
add


Use
BIGGEST
of the two






min
max,
th C th
x x x
E E E

What would be the
minimum
KE of
Products?


think about it!

20

Min. Kinetic Energy
of the products is:



min
th
x
Q E

Energy produced in
the reaction

Minimum required
energy of the incident
particle

What is the threshold for the
7
Li(

,n
)
10
B nuclear reaction?

21

0 MeV
2.06 MeV
4.4 MeV
6.46 MeV
0%
0%
0%
0%
4.4 MeV, 2.06 MeV
th C
x x
E E
 
1.
0
MeV

2.
2.06
MeV

3.
4.4
MeV

4.
6.46
MeV

Remember: Threshold is minimum energy the
incident particle has to have

What is the threshold of
14
C(
p,n
)
14
N ?

And Minimum KE of the Products ?

22

3.1 MeV and 3....
2.1 MeV and 2....
3.1 MeV and 2....
2.1 MeV and 3....
0 MeV and 6 Me...
0%
0%
0%
0%
0%
1
th
x
x
X
m
E Q
m
 
  
 
 

E
x
C

1
.
2
Z
x
Z
X
A
x
1
/
3

A
X
1
/
3
1.
3.1
MeV

and 3.74
MeV


2.
2.1
MeV

and 2.74
MeV


3.
3.1
MeV

and 2.74
MeV


4.
2.1
MeV

and 3.74
MeV


5.
0
MeV

and 6
MeV


14
C(
p,n
)
14
N

23

Q
14.003242
1.007825
1.008665
14.003074
931.494
0.626
MeV
E
x
th
0
E
x
C
1.2
6
1
14
3
1
3
2.11
MeV
Min
.
KE
Prod
2.11
0.626
2.736
MeV
24

Neutron Scattering


First Type:



Scattering

reactions



Elastic

scattering (
Q=0
)



Inelastic

scattering (
Q<0
)


*
X n
n X
X n



 





25

N
Scattering

Kinematics








Solving for the scattering angle:


E
'

1
(
A

1
)
2
E
cos

s

E
(
A
2

1

cos
2

s
)

A
(
A

1
)
Q


2
1
cos ( 1) ( 1)
2
s
E E QA
A A
E E
EE

 

    
 


 
E incident energy
E' final energy
x y n
X Y
m m m
m m M
 
 
*
n X X n
  
Simplifies to:

It is still a binary
rxn
. So eq. still applies, but

atomic mass number
n
m
A
M

What is the energy of a 5MeV neutron after
it is elastically scattered 30
o

by a
10
B atom?

26

1.9 MeV
2.9 MeV
3.9 MeV
4.9 MeV
5.9 MeV
0%
0%
0%
0%
0%

E
'

1
(
A

1
)
2
E
cos

s

E
(
A
2

1

cos
2

s
)

A
(
A

1
)
Q


2
1.
1.9
MeV

2.
2.9
MeV

3.
3.9
MeV

4.
4.9
MeV

5.
5.9
MeV

Hint: for elastic collisions
only “+” matters

Notice it is an elastic collision:

Q=0

27

Accelerator in Columbia (8.4 MeV d)

28

56
56
56
56 3
56
Fe + d +
Fe + d + p
Fe + d + n
Fe + d + He
Fe + d +







How many of these could happen?