5: The Neutron Cycle
B. Rouben
McMaster University
Course EP 4D03/6D03
Nuclear Reactor Analysis
(Reactor Physics)
2015 Sept.-Dec.
2015 September
1
Table of Contents




Reaction rates
Reactor multiplication constant, reactivity, critical
mass
Thermalizing neutrons, Maxwellian distribution
Neutron cycle
2015 September
2
Reaction-Rate Equation





The general equation for a reaction rate must be
stressed, as it is extremely important:
R() = ()()
Remember that the macroscopic cross section 
depends on the type of nuclide (the material), the
type of reaction, and the speed  of the neutrons
relative to the nuclides.
This is a basic equation!
The reaction rate can be integrated over any
range considered for the neutron energies.
Typical units for R are reactions.cm-3.s-1.
2015 September
3
Example: Fission Rate



Let us consider the fission reaction. The fission
cross section is written f. This is a function of
neutron energy E and can be (and usually is) a
function of position r, because there may be
different materials at different points.
Then Fission rate at point r = f(E, r)(E, r).
And the total fission rate in the reactor would be
obtained by integrating this quantity over the
reactor volume.
2015 September
4
Neutron-Production Rate




If the average number of neutrons produced in a fission
is  (don’t confuse this with neutron speed), we can
define a new quantity, the “production” (or “yield”)
cross section f(E, r).
Then
Production rate of neutrons at r = f(E, r)(E, r)
This can also be called the “volumetric source” of
neutrons.
The total neutron production rate in the reactor can be
obtained by integrating the above quantity over r.
It is of course important to distinguish between
fission rate and yield rate (volumetric source).
2015 September
5
Note on Calculating Reaction Rates





To calculate the reaction rates, we need therefore the
macroscopic cross section and the neutron flux.
These are calculated with the help of computer programs:
 The cross sections are calculated from international
databases of microscopic cross sections
 The neutron flux distribution in space (the “flux shape”) is
calculated with specialized computer programs, which solve
equations describing the transport or diffusion of neutrons
[The diffusion equation is an approximation to the more
accurate transport equation.]
Reactor physicists often speak of the “flux shape”, this is the
spatial distribution of the flux.
But remember that to calculate the absolute value of reaction
rates, we should use an absolute flux (not a flux shape).
The absolute value of the neutron flux must be tied to a
measurable quantity, e.g., the total reactor power.
2015 September
6
Reactor Multiplication Constant



Several processes compete for neutrons in a nuclear
reactor:
 “productive” absorptions, which end in fission
 “non-productive” absorptions (in fuel or in structural
material), which do not end in fission
 leakage out of the reactor
Self-sustainability of the chain reaction depends on
relative rates of production and loss of neutrons.
The self-sustainability of the fission reaction in the finite
reactor is measured by the reactor multiplication
constant:
Rate of neutron production
keff 
2015 September
Rate of neutron loss (absorptions  leakage)
7
Reactor Multiplication Constant


Another common definition of keff is: ratio of number of
neutrons born in one “generation” to number born in the
previous generation. It can be shown that the two
definitions of keff are equivalent.
Three possibilities for keff :
 keff < 1: Fewer neutrons being produced than lost.
Chain reaction not self-sustaining, reactor
eventually shuts down. Reactor is subcritical.
 keff = 1: Neutrons produced at same rate as lost.
Chain reaction exactly self-sustaining, reactor
in steady state. Reactor is critical.
 keff > 1: More neutrons being produced than lost.
Chain reaction more than self-sustaining,
reactor power increases. Reactor is supercritical.
2015 September
8
Critical Mass





Because leakage of neutrons out of the reactor increases as
the reactor size decreases, the reactor must have a minimum
size for criticality.
Below minimum size (critical mass), leakage is too high and
keff cannot possibly be equal to 1.
Critical mass depends on:
 shape of the reactor
 composition of the fuel
 other materials in the reactor.
Everything else being constant, the shape with lowest relative
leakage, i.e. for which critical mass is least, is the shape with
smallest surface-to-volume ratio: a sphere.
All these concepts will be made more concrete as we go
along in the course.
2015 September
9
Reactivity

Reactivity (r) is a quantity closely related to reactor
multiplication constant. It is defined as
1
r = 1
keff

= (Neutron production - loss)/Production
= Net relative neutron production
“Central” value is 0:




r < 0 : reactor subcritical
r = 0 : reactor critical
r > 0 : reactor supercritical
Reactivity is also often referred to as an increment,
e.g., adding positive or negative reactivity to a
reactor, i.e., increasing or decreasing r.
2015 September
10
Units





The multiplication constant keff is a ratio of like
quantities, therefore it is a pure number and has no units.
Similarly, the reactivity r is also a pure number, and has
no units.
However, because in reactor physics we seldom deal
with k values extremely different from 1, reactivity is
often written in “units” of a small fraction of unity. Two
of the common units are
 1 milli-k (or mk)  0.001
-5
 1 pcm  10 = 0.01 mk
Thus, e.g., a reactivity of +3 mk means r = +0.003, and
a reactivity of -50 mk means r = -0.050
1 mk may seem small, but one must consider the time
scale
on which the chain reaction operates.
2015 September
11
Fission Neutrons


Neutrons from fission
have a distribution of
energies, mostly in the
1-to-a-few-MeV range.
This distribution has a
maximum at ~1 MeV
(neutron speed ~13,800
km/s!).
Energy Distribution of Fission Neutrons
2015 September
12
Fast and Thermal Neutrons
• Neutrons may lose energy by collisions with
materials in the reactor, i.e., they may be slowed
down.
• Maximum slowing down is to a distribution of
energies in thermal equilibrium with the ambient
environment.
• For a temperature of 20o C, these “thermal”
energies are of order of 0.025 eV, i.e. neutron
speed = 2.2 km/s, orders of magnitude slower than
fast neutrons, but still ~ speed of bullet.
2015 September
13
Why Thermalize Neutrons?


Fissile nuclides are nuclides which can be
fissioned by neutrons of any energy, and are
therefore easier to fission. 235U, 239Pu and 241Pu
are fissile and are the main fissioning nuclides in
most reactors [although 235U is in small
abundance (0.72% of natural uranium)], and 239Pu
is created in the reactor].
The probability of a neutron inducing fission in a
fissile nuclide is very much greater (by orders of
magnitude) for very slow neutrons than for fast
neutrons (see next Figure).
2015 September
14
Schematic View of a Typical Cross Section,
Showing Resonances
2015 September
15
Thermal Reactors





Thermal reactors are designed to take advantage of the
much greater probability of inducing fission of fissile
nuclides at low neutron speeds.
Therefore, a moderator is used to thermalize neutrons.
However, as neutrons slow down from ~MeV energies to
thermal energies, they may be absorbed in fuel.
In the “resonance” energy range, ~ 1 eV - 0.1 MeV, the
probability of non-productive capture in fuel is great.
Capture resonances “rob” neutrons from the chain
reaction.
Therefore, ways are sought to minimize resonance
capture. Lumping the fuel into channels separates the
moderator from the fuel and helps to reduce resonance
capture.
2015 September
16
Speed Distribution in a Population of Particles




In a population of free particles, there will be a
distribution of particle speeds (or energies).
The temperature of a gas is a measure of the
energy of motion of the gas molecules.
The molecules are flying around at various
speeds, and exchanging energy in collisions.
Maxwell showed that in a gas at steady
temperature T (on the absolute, Kelvin scale), the
molecules will have a distribution of speeds
given by a function which he derived.
2015 September
cont’d
17
The Maxwellian Distribution of Speeds


Let m be the mass of the particles and T (K) be
the absolute temperature.
If we write the fraction of particles with speed in
an interval dv about speed v as n(v)dv, then the
Maxwellian distribution of speeds is:
3 / 2
mv 2

2
2 kT
4  2kT 
n(v)dv 
dv ( see graph next slide )

 ve
 m 
where k  Boltzmann cons tan t  1.380658 *10  23 J / K
( Exercise : Show that
2015 September

 nv )dv  1)
0
18
Maxwellian Distribution in v
n(v)
n(v)
0
1000
2015 September
2000
3000
v (m/s)
4000
5000
6000
7000
8000
9000
19
The Maxwellian Distribution of Energies

The kinetic energy E of a particle is related to its
speed  by E = m 2/2. The Maxwellian
distribution can therefore be (and often is)
written in terms of E instead of v.
If we write n( E )dE  n(v)dv, we can derive (exercise !)
d
n( )
n( )
2
kT )3 / 2 E 1 / 2 e  E / kT
n( E )  n( )



dE dE / d
m

(See the graph on the next slide)
2015 September
20
Maxwellian Distribution in E
n(E)
n(E)
0.00E+00
5.00E-02
1.00E-01
1.50E-01
2.00E-01
E (eV)
2015 September
21
Maxwellian Distribution for Thermal Neutrons


In a nuclear reactor, neutrons are “thermalized”
by the moderator, i.e., they are slowed down to
the “thermal” energy range (<~ 0.6 eV), where
they are in thermal equilibrium with the ambient
environment.
As a consequence, thermal neutrons in a reactor
are almost in a Maxwellian distribution – the
distribution is slightly perturbed by the
absorption of neutrons in the fuel.
2015 September
22
Exercises in Thermal-Neutron Distribution
Exercises:
 From the form of the Maxwellian in v and in E,
show that the most probable kinetic energy in the
Maxwellian distribution is
Epeak=kT/2 (≈ 0.0125 eV at room temperature),
 Show on the other hand that the peak speed
vpeak corresponds to an energy of kT (≈0.025 eV at
room temperature), and that this gives a value
vpeak = 2,200 m/s (not that slow, faster than a bullet)
2015 September
23
Neutron Cycle


The next slide illustrates the cycle of
neutrons in fission, slowing down,
absorptions and leakage in a thermal
reactor.
We will come back to this cycle more
quantitatively as we go on in the course.
2015 September
24
Neutron Cycle in Thermal Reactor
Neutrons Born in Fission
Fast-Neutron Leakage
Fast Fissions
Neutrons Slowing Down
Neutrons Captured in
Fuel Resonances
Thermal-Neutron Leakage
2015 September
Non-Productive ThermalNeutron Absorptions in
Fuel & Other Materials
Thermal Fissions
25
Exercise


From anything you knew previously, or anything
you think you knew from general knowledge, or
anything that you have learned from this learning
module:
List as many things which are important in
running a nuclear plant as you can think of, and
for each item on the list indicate whether that
item is closely, not so closely, or not at all,
related to reactor physics.
2015 September
26
END
2015 September
27