Principles of nuclear energy

Introduction
Principles of Nuclear Energy

P
e-
X
A
Z
where:
X is the element symbol
A is the # of protons plus neutrons
Z is the # of protons only
Bohr’s Model
U
235
92
EXAMPLE
Nuclear Energy (Basics: Atomic Structure)
• Z = atomic number
– number of protons in an atom
• N = number of neutrons
• A = Z + N = mass number
– number of protons plus number of neutrons

P
e-
N
e-
P N
e-
P
N
HYDROGEN DEUTERIUM TRITIUM
These are what of Hydrogen?

• Two nuclei with the same number of protons can
have different number of neutrons and are called…
– isotopes of the same element.
• Example:
1
3
1
2
1
1
Z
A
H
H
H
H
isotopes
+ + +
HYDROGEN DEUTERIUM TRITIUM
:

• Atomic Mass Unit is a unit of mass equal to
1.66x10-27 kg.
• Proton (+1) 1.007277 amu
• Neutron (0) 1.008665 amu
• Electron (-1) 0.000548 amu
• Positron (e+, b+) – Positively charged electron
• Neutrino (u) – Electrically neutral. Do not
react.

Nuclear Energy (Nuclear Equations)
In balancing nuclear eqns., the same nucleons show up in the products
as entered the reaction. e.g.
4
4
3
3
2
2
1
1
A
Z
A
Z
A
Z
A
Z N
M
L
K 


To balance, the following relationship must be satisfied:
4
3
2
1
4
3
2
1 ; A
A
A
A
Z
Z
Z
Z 





Sometimes the symbols γ or ν are added to products to indicate
emission of electromagnetic radiation or a neutrino. They have no
effect on balancing as both have zero Z and A.
Reactions are either exothermic or endothermic.

EXAMPLE
An exothermic reaction occurs when common aluminum is
bombarded with high-energy α particles, resulting in Si30 (a heavy
isotope of silicon). In the reaction, a small particle is emitted. Write
the complete reaction and calculate the change in mass.
Nuclear Energy (Nuclear Equations)

Nuclear Energy (Energy from Nuclear Reactions)
Lets have a closer look at the release of energy by nuclear means.
We find that the energy yield from a kg of nuclear fuel is more than a
million times that from chemical fuel… a million??!!!
Proof ????
E = (1 kg)(3x108 m/s)2 = 9x1016 J
Nuclear fission is relatively inefficient, since the ‘burning’ of 1 kg of
uranium involves the conversion of only 0.87 g of matter into energy.
This corresponds to about 7.83x1013 J/kg of the uranium consumed.
The energy of combustion of a familiar fuel like gasoline is 5x107 J/kg.
The ratio of these, 1.57x106, reveals the tremendous difference between
nuclear and chemical energies.

Nuclear Energy (Energy from Nuclear Reactions)
Energy corresponding to change in mass in a nuclear reaction is
calculated from Einstein’s law: E = mc2
As it is convenient to express masses of nuclei in amu and energy in
Joules and MeV, therefore,
)
(
10
49
.
1
)
( 10
amu
in
m
x
J
in
E 

 
and
)
(
931
)
( amu
in
m
MeV
in
E 


Therefore, the exothermic reaction in the previous example produces
-0.00254 x 931 = -2.365 MeV of energy.
Note: See Table 9-1 for other mass-energy conversion factors.

• FISSION
– a nucleus breaks up into smaller parts
– used in nuclear power plants (Uranium-235, Plutonium-239, Uranium-233)
Following are reasons why not all the fission neutrons cause further
fission:
1. Non-fission capture or absorption of some neutrons by the
fission products, non-fissionable nuclei in the fuel, structural
material, coolant, moderator and so on.
2. Leakage of neutrons escaping from the core.
• FUSION
– lighter nuclei come together to build heavy nuclei
– important in stellar energy generation
– Typical Temperature for Fusion = tens of millions of degrees
Kelvin.
Nuclear Energy (Fission and Fusion)

• In a star, its basic source of energy is the conversion of hydrogen
into helium.
)
(Positrons
2
+
Atom)
(Helium
1
fusion
Atoms)
(Hydrogen 

 

4
Nuclear Energy (Fission and Fusion)
Note: The process is so difficult to control that it is questionable
whether commercial adaptation will ever be economically feasible.
Fusion reactors would be fueled by deuterium available in almost
unlimited supply in sea water.
[See Table 9-2 (page 363) for possible fusion reactions]
Decrease in mass is about 0.0276 amu and, thus, produces
-0.0276 x 931 = -25.7 MeV.
Heat produced triggers/sustains succeeding reactions.

235U + n  236U*  (A1,Z1) + (A2,Z2) + Nn + E
• Z1 + Z2 = 92, A1 +A2 + N = 236
• A1 =A2, symmetric fission rare (~0.01%)
• Capture of neutron by 235U forms compound nucleus(*)
• ~2.4 (on average) prompt neutrons released per fission event
• Immediate products are called fission fragments. They, and
their decayed products, are called fission products.
Nuclear Energy (Basics: Fission Equation)
235U + 1n  236U*  137Ba + 97Kr + 2 1n + E
92 0 92 56 36 0

Symmetric Fission
Nuclear Energy (Basics: Fission Yield)
75 ≤ A ≤ 160
The probability that a
particular pair of
fission fragments will
be produced by fission
Most probable fission product
ranges:
and
85 ≤ A ≤ 105
130 ≤ A ≤ 150

MASS OF REACTANTS
U: 235.0439 amu
n: 1.00867 amu
--------------------------------
Total : 236.05257 amu
--------------------------------
MASS OF PRODUCTS
Ba: 136.9061 amu
Kr: 96.9212 amu
2 n: 2(1.00867) amu
--------------------------------
Total: 235.84464 amu
--------------------------------
m = 235.84464 - 236.05257 = -0.20793 amu = -193.583 MeV
Nuclear Energy (Energy Released)
For the reaction,
235U + 1n 137Ba + 97Kr + 2 1n
92 0 56 36 0
which is the same for U-233 and Pu-239.






energy
prompt
More energy is, however released due to (i) slow decay of the fission
fragments, and (ii) non-fission capture of excess neutrons in reactions that
produce energy, though much less than that of fission.

The total energy produced per fission reaction is about 200 MeV.
The complete fission of 1g of U-235 nuclei thus produces,
Nuclear Energy (Energy Released)
Avagadro’s Number x 200 MeV = 0.60225x1024 x 200
U-235 mass 235.0439
= 0.513x1024 MeV
= 2.276x1024 kWh
= 8.19x1010 J
= 0.948 MW-day
Fuel burnup: The amount of energy in MW-days produced of
each metric ton of fuel.
Fuel: All uranium, plutonium and thorium isotopes. Does not
include other compounds or mixtures. Fuel material refers to
fuel plus such other material.

• The total energy from fission after all of the particles from decay
have been released, is about 200 MeV.
Nuclear Energy (Basics: Energy from Fission)
MeV
Fission fragment kinetic energy 166
Neutrons 5
Prompt gamma rays 7
Fission product gamma rays 7
Beta particles 5
Neutrinos 10
Total 200

• Most of the naturally occurring isotopes are stable.
• Some isotopes of heavy elements (Z = 81-83) and all the
isotopes of the heavier elements beginning with Polonium (Z =
84) are unstable (or radioactive) i.e. binding energy per nucleon
is small.
• Natural and artificial radioactive isotopes are also called
radioisotopes.
• A spontaneous disintegration process called radioactive decay
occurs. The resulting nucleus is called the daughter and the
original nucleus is called the parent. Daughter may or may not
be stable.
• Radioactivity is always accompanied by a decrease in mass. The
energy thus liberated shows up as KE of the emitted particles
and as electromagnetic radiation (γ-rays).
• Isotopes decay by spontaneous emission of alpha, beta, or
gamma rays. Artificial isotopes also undergo Positron decay,
K-capture and also emits neutrons and neutrinos.
Nuclear Energy (Radioactivity)

• Alpha Decay (4α+2)
– Commonly emitted (at 0.1 times the speed of light) by heavier radioactive
nuclei.
– Quickly ionized to stable He atom
– Penetrating power is low
– Example: 94Pu239 → 92U235 + 2He4
• Beta Decay (0β-)
– Usually accompanied by the emission of neutrino and gamma rays.
– Penetrating power of Beta particles is small compared to gamma rays but is
larger than that of alpha particles. (0.9 times the speed of light )
– Can present a biological hazard w/o proper shielding
– Example: 82Pb214 → 83Sn214 + -1e0 + n
• Gamma Radiation (γ)
– Electromagnetic radiation of high frequency and, thus, high energy (e = h.n)
– No charge and no mass (like an X-ray) and great penetrating ability
– By emitting γ-radiation, an excited daughter nucleus falls back into its stable
ground (lowest) energy state.
– Biohazard to nuclear reactor operators: must be shielded (water - good, steel
- better, lead - best)
– Example: [ ZXA ]* → ZXA + γ
2

• Positron Decay (+1e0)
– Positrons have very low penetrating power.
– Occurs when radioactive nucleus contains an excess of protons
– Electron is also released to maintain neutrality (!!!annihilation!!!)
– Reverse of annihilation is called pair production.
– Example: 15P30 → 14Si30 + +1e0 ; -1e0 + +1e0 → γ (E = 1.024 MeV)
• K - capture
– When a nucleus has excess of protons but not threshold energy to emit
positron, it captures an electron from the K-shell.
– Accompanied by X-ray emission.
– See Figure 9-6 (on pp. 369) of your book.
– Example: 29Cu64 + -1e0 → 28Ni64
• Neutron emission
– May occur when nucleus possesses high excitation energy.
– The daughter (54Xe136) is an isotope of the parent (54Xe137).
– Occurs rarely, however, it happens in nuclear reactors and hence, is a
source of delayed fission neutrons.
– Example: 54Xe137 → 54Xe136 + 0n1
2

• The emission of radiation from the nucleus of an atom is random but if the
number is large, a rate of decay can be established.
• The rate of decay is directly proportional to the number of radioactive nuclei,
N, that the sample contains. The rate at which one-half of the sample decays is
called the half-life (q1/2 or tH).
• Half-life can range from very small fractions of a second to billions of years.
• This rate of disintegration is independent of the external conditions such as
temperature and pressure.
• At any time q (or t), the ratio of number
of nuclei present (N) to the initial number
(N0) is given by
2
Nuclear Energy (Decay rates & Half-lives)
2
/
1
/
0 2
1
q
q







N
N

If N is number of radioactive nuclei at any time q and dN is the number decaying in
an increment of time dq , the rate of decay is directly proportional to N.
(l is decay constant. Units: s -1)
The rate of decay (lN) is also called activity (A) and has dimensions disintegrations
per second (dis/s) or simply s-1. Thus
2
lq
l
q




e
N
N
get
we
g
Integratin
N
d
dN
0
,
lq

 e
A
A 0
In terms of half-life:
Done by measuring the change in activity with time and computing l from the
slope of the activity history on a semi-log plot. See Figure 9-8 (on pp. 373) of your
book.
l
l
q
lq
6931
.
0
2
ln
2
1
2
/
1
0
0
2
/
1




 
or
e
A
A
N
N
Also see Figure 9-7 (on
pp. 372) of your book.

EXAMPLE
Radium 226 decays into radon gas. Calculate (a) the decay constant
and (b) the initial activity of 1g of radium 226 (atomic mass is
226.0245 amu).

• Kinetic energy of a neutron is given by
Since mn = 1.008665 amu, then
En (MeV) = 5.227x10-19 V2
where V is in cm/s.
• Newly born fission neutrons have energies of 0.075-17 MeV.
• They travel, collide with other nuclei and get slowed down. This process
is called scattering.
• Neutrons are classified into 3 categories according to their energy (in eV)
as fast (> 105), intermediate (1-105) and slow (<1).
• Newly born fission neutrons carry, on average, ~2% of a reactor fission
energy. They are either prompt or delayed.
• Delayed neutron energies play a vital role in nuclear reactor control.
• Most prompt neutrons have energies less than 1 MeV but average around
2 MeV (See Figure 9-9 (on pp. 375) of your book)
Nuclear Energy (Neutron Energies)
2
2
1
V
m
E
KE n
n
n 


• Lowest energy state that can be attained by neutrons is to be in thermal
equilibrium with the molecules of the medium they are in. In this state, they
become thermalized and are called thermal neutrons.
• Neutrons possess a wide range of energies and corresponding speeds.
• The most probable velocity of a neutron is the one that corresponds to the
maximum number density and is given by:
where K is Boltzmann’s constant = 8.617x10-11 MeV/K. For the known mass of the
neutron, we get Vm = 128.4 T 1/2 (m/s)
• The energy corresponding to the most probable velocity is:
and for the known mass of the neutron, we get
Em = 8.617x10-5 T (eV)
• The energy of the thermalized particle is independent of mass and only a
function of the (absolute) temperature of the medium.
Nuclear Energy (Thermal Neutrons)
m
KT
Vm
2

KT
mV
E m
m 
 2
2
1

• When neutrons become thermalized, they possess the same energy
distribution as the molecules of the medium.
• The velocities are dependent on mass.
• At 20 °C, Vm = 2198 m/s 2200 m/s and Em = 0.02524 eV. (See
Table 9-5 (on pp. 378) of your book). This is sometimes considered
as ‘standard’. Cross-sections (See next slide) for thermal neutrons
are customarily tabulated for 2200 m/s neutrons.
• Neutrons having energies greater than thermal such as those in the
process of slowing down in a thermal reactor, are called epithermal
neutrons.
Nuclear Energy (Thermal Neutrons)


• Interaction rate between a beam of neutrons and nuclei in a target material is
proportional to (i) neutron flux (f) (ii) number of atoms in the target.
• Consider a beam of monoenergetic (same KE) neutrons of speed V cm/s and
density n neutrons/cm3 incident on a target of A cm2, thickness dx and containing
N nuclei/cm3.
f = nV ; Interaction rate = s f (N.A.dx)
s is called the microscopic cross-section or the cross-section of the reaction. σ
has the units cm2/nucleus. It can be regarded as the area presented by each
nucleus to neutrons to cause a reaction.
• The radius rc of a nucleus is given by:
rc = 1.4 x 10-13 A1/3 (A is mass number.)
• cm2 is too large a unit. The cross-sectional area of an average nucleus is taken as
the unit of  called barn. Thus, 10-24 cm2 = 1 barn. Cross-sectional values vary
from small fractions of a millibarn to several thousand barns.
• Neutrons have as many cross sections as there are reactions. The reactions can
be scattering (s), absorption (a), capture (c) and fission (f). Total cross-
section (t) is the sum of these cross-sections (See Figure 9-12 to 9-14 (on pp.
381-383) of your book).
sa = sc + sf
Nuclear Energy (Cross-section)

*graph courtesy of World Nuclear Association

• The product sN is equal to the total cross-section of all
the nuclei present in a unit volume and is called
macroscopic cross-section (Σ) and has units cm2/cm3 or
cm-1. It can also be explained as the probability per unit
length that a neutron will collide, i.e. the collision cross-
section.
• Macroscopic cross-sections are also designated according
to the reaction they represent. etc.
• λ = 1/Σ = mean free path. Represents the average distance
that a neutron travels without making a collision or
interaction with a target nucleus.
• For an element of atomic mass (A) and density r (g/cm3),
N (nuclei/cm3) can be calculated from
A
number
s
Avagadro
N
'
r

f
f
s
s N
N s
s 


 ,

• Neutron cross-section for any nucleus depends upon the energy of the neutron
reacting with it.
• σt σa (σs << σa) Also, for most nuclei, σs does not change much with
neutron energy En. (See Figures 9-12 to 9-14 (on pp. 381-383) of your book).
• Variation of absorption cross-sections with neutron energy can be divided into 3
regions: (1) 1/V region, (2) resonance region and (3) fast neutron region.
Nuclear Energy (Variations of Cross-section)

n
a E
C 1
1

s
(  V
C
V
m
C
n
a
1
2
1
2
2
/
1
2
1 







s
1. 1/V region
In the low-energy region, σa is inversely proportional to the square root of the
neutron energy En. i.e.
Thus,
where C1 and C2 are constants, V is the neutron velocity.
What does this indicate??

The 1/V law may also be written as
where the subscripts 1 and 2 refer to two different neutron energies within the
1/V range.
The upper limit of this region is different for different nuclei.
n
n
a
a
E
E
V
V 2
,
1
2
2
,
1
,


s
s
*graph courtesy of World Nuclear Association

2. Resonance region
• Most neutron absorbers exhibit one or more peaks occurring at definite
neutron energies, called resonance peaks. They affect neutrons in the
process of slowing down.
• U-238 has very high resonance absorption cross sections, with the highest
peak, at about 4000 barns occurring at about 7eV. This fact affects the
design of thermal reactors because U-238 absorbs many of the neutrons
passing through the region & affects the reactor neutron balance.

3. Fast-neutron region
• As the neutron energies increase beyond the resonance region, the
absorption cross-sections gradually decrease. At very high values of En
3
/
2
2
0
2
2
2 A
r
rc
t 

s 

barns
A
t
3
/
2
125
.
0

s


 s
a
t s
s
s 2 x cross-sectional area of the target nucleus
i.e.
Using r0 = 1.4x10-13 cm and 1 barn = 10-24 cm2, we get
• In very high neutron
energy range, total cross
section (st) is very low,
usually less than 5 barns
for heavier nuclei.

• A moderator is used to slow down the neutron in a reactor.
• Neutrons must be slowed by elastic collisions.
• An effective moderator is one that reduces neutron energy with relatively
few collisions.
• The average energy lost per elastic collision is expressed in terms of a
quantity called logarithmic energy decrement, ξ , and is given by
where A is the mass number of struck nucleus (moderator).
• Moderating ratio (Should be as large as possible)
• Moderating power (Should be as large as possible)
• Moderating ratio is a relative measure of the ability of a moderator to
scatter neutrons without appreciably absorbing them.
Nuclear Energy (Moderators)











1
1
ln
2
)
1
(
1
2
A
A
A
A

a
s
a
s



 
s
s

s
s
N 

 
s


• Surrounds the core, which is a medium of low
neutron absorption and high neutron scattering
cross-section.
• Reduces neutron leakage and, thus, required fuel
mass.
• Moderators like H2O, D2O etc. are suitable
reflectors for thermal reactors.
• Fast neutrons are reflected by heavier materials like
natural uranium (i.e. U-238 that also serves as
source for Pu-239).
• Bare cores have serious thermal distribution
problems. Reflectors improve the neutron flux
distribution of a reactor core by flattening it within
the core.
Nuclear Energy (Reflectors)

Principles of nuclear energy

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