Nuclear Fundamentals
1.) Definitions
α,β,γ,ν,n, p
2.) Mass Energy Relationships
Particles;
Eo = m oc 2
E o is the energy equivalent to rest mass
(m o ) of particle,
c = speed of light, ≈ 3 x 108 m/s
Photons
E = hν
h = Planck’s constant = 6.626 x 10-34 joule-s
ν = frequency of radiation, s-1
deBroglie equation,
h
h
=
p mv
hc
Å
=
= 12. 4
E
E (keV )
λ =
3.) Radioactive Decay of Unstable Isotopes, A X Z
Particles
X Z → A − 4 X Z −2 + 4 α 2
(parent)
(daughter)
A
238
Pu 94 → 234 U 92 + 4 He 2
From nuclear reactor physics we find;
Eα =
E
=
Total
transitional
energy
E

 4. 0026  
1
+



A


d

Qa +
∆E
between
ground
state of
parent &
daughter
Ep
-
Excitation
energy of α
emitting
level in
parent
Ed
Excitation
of daughter
nucleus
fed by decay
=========================================
Electron Events
Too many n’s
n o →1p1 + 0β −1 + ν e
1
A
X Z → A Y Z +1 + 0β −1 + ν e
Too few n’s
1
p1→1n 0 + 0β +1 + ν e
A
X Z → A Y Z −1 + 0β +1 + ν e
Too few n’s
1
p1 + 0β +1→1n 0 + ν e
A
X Z + 0β −1→ A Y Z −1 + ν e
---------------------------------------------Example 238 Pu 94 decay chain
4.) Law of Radioactive Decay
dN
= −λN
dt
N= # of radioactive atoms
λ = decay constant =
t 1 = half life
0. 693
t1
2
2
1 Curie = 3.7 x 1010 disintegrations per second
====================================
Can rewrite decay equation:
∫ dN/N = ∫ λ dt
which gives
N = No exp (-λt)
Can also express the instantaneous power released:
Q (t) = Qo exp (-λt)
Problem
Assume a radioisotope system which contains 1 kg of
PuO2 (90% 238Pu) which initially operates at 0.4 W/g of
238Pu. What is the thermal power after 10 years?
A.) t 1 = 87.75 y
2
0.693
λ =87.75 = 0.0079 y-1
B.)
0.9 238Pu
238 g Pu
)}
Qo={1000 g PuO2 • (270 g PuO ) •(
Pu
2
0.4 W
•[ g 238 Pu ]
= 317 Watts
C.) After 10 years;
Q = 317 • exp (-0.0079 x 10)
= 293 Watts
5.) Cross Sections
Interaction probability for a given reaction by a
particle traveling through a given medium;
microscopic, σ cm2
so,
macroscopic, Σ = N σ
(1 barn = 10-24 cm2)
cm-1
Σ dx = probability of interaction
per unit track length
Removal of particles from a given state;
Or,
-dΦ(x) = Φ (x) Σ dx
Φ (x) = Φo exp (-Σx)
============================
Mean distance before undergoing a reaction:
λ = 1/Σ
============================================
6.) Neutron Spectra
When neutrons, which are emitted during the fission
process, slow down and are thermalized, they come close to
a Maxwell -Boltzmann distribution.


dn  2 π n 
E
n(E ) =
)• E
•
exp(−

3
dE 
kT

π
kT
(
)
2


Figures 2.12 & 2.13
@ T= 20°C most probable thermal velocity is 2200 m/s
which give E = 0.0252 eV
For;
En < 0.0252 eV
1<En< 100 eV
0.1<En< 1 keV
0.5<En< 15 MeV
cold neutrons
resonance n’s
intermediate n’s
fast neutrons
Lots of Cross Sections!
Figure 2.14
=======================================
7.) Neutron Slowing Down
Elastic scattering useful for dropping the n energy from the
MeV range to the eV range where fissioning is more
probable (σfiss is the highest)
(A-1)
Emin = [(A+1) ]2 Eo = αEo
A = At. mass of nuclide/At. mass of neutron
-----------------------------------------------Logarithmic Energy Decrement
ξ =ln (
Εο
Ε
)= 1 +
(α•ln α)
(1-α)
Would like high values of ξ
See Table 2.3
=========================================
8.) Production of Isotopes
Formation rate = Φσ act N − λ N
where:
σ act is cross section for producing isotope Y from parent X
N = no. of parent X atoms per cc.
However, if isotope Y is radioactive, then N* is the number
of Y atoms per cc at any time.
decay rate = λ N*
Net production of Y:
dN *
= Φσ act N − λ N
dt
Can show that,
 Φσ act N  
• [1 − exp(− λ t r )]
N* = 



λ


Activity after the neutrons are turned off:
[
]
λ N * = Φσ act N • 1 − exp(− λ t ) • exp(− λ t r )
Where tr = time after removal from the n flux
=======================================
9.) Some Observations on Reactor Operation
# of fissions (or n’s) in one generation
Define k = ___________________________
# of fissions (or n’s) in the
immediately preceding generation
k<1 subcritical
k =1 critical
k> 1 supercritical
Figure 3.2
Description of space reactor - Figure 3.3
--------------------------------------------Fissionable elements-Tables 3.1 & 3.2
-------------------------------Energy released - Table 3.3
----------------------------------Fission Products -Figure 3.4