227.

00-5

Nuclear Theory - Course 227

NEUTRON MULTIPLICATION FACTOR AND REACTIVITY

In the previous lesson the neutron multiplication factor

(k) was defined as:

no. of neutrons in one generation

k = no. of neutrons in the preceeding generation

This definition is only valid if the neutron power is high

enough that the effect of source neutrons (photoneutrons and
spontaneous fission neutrons) may be ignored and if k itself
is not changing. A more precise way to define k is as the
product of six factors, each of which represents a possible
fate for a member of the neutron population. Thus:

Where:

£ (epsilon) = Fast Fission Factor. The factor by which the

fast neutrons population increases due to fast
fission.

No. of neutrons from No. of neutrons from

£ = thermal fission + fast fission
No. of neutrons from thermal fission

A typical value is about 1.03 for natural uranium fuel

p = Resonance Escape Probability. The probability

that a neutron will not undergo resonance
capture in U-238 while slowing down.

No. of neutrons leaving

resonance energy range
p = No. of neutrons entering
the resonance energy range

July 1979

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 LLI.UU-~

A typical value is about 0.9 for natural uranium fuel.

n (eta) = Reproduction Factor. The number of neutrons

produced by thermal fission per neutron absorbed
by the fuel.

fuel I fuel
I
n = V f = V f
I fuel fuel
L fuel
a I f + D,Y

A typical value is about 1.2 for natural uranium fuel.

f = Thermal utilization. The fraction of the thermal

neutrons absorbed by the fuel compared to neutrons
absorbed in the whole reactor.

L fuel
f = a
L total reactor
a

A typical value is about 0.95 for a CANDU reactor core.

Note: Fuel must be defined the same way for both n & f.

Af = Fast Non-leakage Probability. The probability

that a fast neutron won't leak out of the
reactor. A typical value is about 0.995.

At = Thermal Non-leakage probability. The probability

that a thermal neutron won't leak out of the
reactor. A typical value is about 0.98.

The first four factors, which depend only on the materials

of construction, are frequently grouped together and called the
multiplication factor for an infinite reactor (k ro ) .

This is normally referred to as the "four-factor formula".

The last two factors are leakage factors which depend on

the size and shape (ie, the geometry) of the reactor. Figure 1
shows how each of the factor relates to the neutron life cycle.

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 227.00-5

Radiative
Absorption J---~~
capture
in
Fuel
Fission

Thermal Neutrons Available

to be Absorbed by the fuel
in the i th generation

k = N£pnfAfAt = EpnfAfAt
N

Fast
Fission
Thermal neutrons available
to be absorbed by the fuel
in the (i+l)th generation

Parasitic
Absorption
n non -
uel reac-
or mater-
'also

Thermal Resonance
Leakage Capture

Neutron Life Cycle

Figure 1

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 227.00-5

A thermal neutron which is absorbed by the fuel may be

absorbed by fissile material (U-235 or Pu-239) or by non-
fissile material (Fission Products, U-238, etc). If it is
absorbed by fissile material it may undergo radiative capture
or cause fission. If it causes fission, v fast neutrons will
be produced. The reproduction factor (n) accounts for all
of this. Thus for N thermal neutrons absorbed by the fuel
Nn fast neutrons are produced.

As U-238, Pu-239, and U-235 all have small but finite

fission cross-sections for fast neutrons, the fast neutrons
can cause additional fissioning to take place. This results in
an increase (~3%) in the fast neutron population. The fast
fission factor (E) accounts for this increase in the fast neutron
population. Thus for Nn fast neutrons from thermal fission we
get ENn fast neutrons from fast and thermal fission.

The two factors E and n are essentially properties of the

fuel and the magnitude of the product En fixes the tolerable
limits of the other factors which can be rAgarded as design
variables. That is the product pfhfh ~ _l_
t
En.
While slowing down the fast neutrons may reach the boundary
of the reactor and leak out. To account for this reduction in
the population we have the fast non-leakage probability (hf).

The fast neutrons may also suffer resonance capture while

slowing through the resonance energy range. The resonance escape
probability (p) accounts for this. Thus for Nsn fast neutrons
starting the slowing down process Nsnphf neutrons reach thermal
energy.

A certain percentage of the thermal neutron population will

diffuse to the boundary and leak out. We use the thermal non-
leakage probability (h th ) to account for this loss.

The remaining thermal neutrons will either be absorbed by

the fuel or by the core material. The thermal utilization factor
(f) accounts for this. Thus for NEpnfhfht thermal neutrons,
NEpnfhfht are absorbed by the fuel.

From Figure 1 you can see that if we divide the number pf

neutrons in the (i + l)th generation by the number in the i th
generation we have: N fA A
k = Epn f t
N = Epnfhfht
When k = 1 the reactor is said to be critical. If k is unity
and the effects of source neutrons are negligible, neutron power will
be constant in a critical reactor. It is important to realize that
a reactor may be critical at any power level and that telling some-
one that a reactor is critical tells them nothing about the reactors
power output. By analogy; if I tell you that a car is not acceler-
ating, do you know how fast it is going?

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 227.00-5

If we want to increase power we must make k greater than

one by reducing the losses, with respect to fission, of neutrons.
The reactor is then said to be supercritical. Power will continue
to increase as long as k is maintained at a value greater than
one.

To reduce power we must increase the losses of neutrons

thus making k less than one. The reactor then is said to be
subcritical and power will decrease until the source neutrons
become significant. (This point will be covered in detail in
lesson 227.00-9.)

Reactivity

A reactor is critical when k = 1. The factor that

determines how subcritical or supercritical a reactor may be,
is the amount by which k differs from 1.

A quantity called reactivity, is used to describe changes

in k which are called reactivity changes. Reactivity is
defined as:

k - 1
k

For values of k close to 1 (eg, 0.98 to 1.02) which easily

encompasses our normal operating range.

Reactivity may be approximated as

6k = k-l
This is the accepted meaning of reactivity in Hydro.

The reactivity changes that are made for normal reactor

control are always quite small, and they are measured in a
unit called the milli-k or mk. (This is not strictly a unit
but is a fraction, 1 mk is the same as 0.1%, ie, 0.001).

For Example: k = 1.002

6k = k - 1

= 1.002 - 1

= 0.002 or 2 mk

A typical CANDU reactivity control system such as the liquid

control zone at Bruce and Pickering have a range of about 6 mk.

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 227.00-5

ASSIGNMENT

1. Put your text and your notes away. Now, write the six
factor formula, define each of the terms, and sketch the
neutron life cycle with the terms used correctly.

2. Calculate the exact value of reactivity for k = 0.95.

3. Calculate each of the six factors for the neutron life

cycle shown below.

415
Undergo Radiative
Capture

816 401
Thermal Neutrons ...._ _~-~ Cause Fission
Absorbed by the *'
Fuel

50 975
Thermal Neutrons Fast Neutrons
Absorbed by Non- From Thermal
Fuel Materials Fission

25
Fast Neutrons
33 From Fast
Thermal Neutrons ~ Fission
Leak Out

899 1000
Neutrons are Fast Neutrons
Thermalized from Fast & Ther-
mal Fission

94 7
- 6 - Fast Neutron Fast Neutrons
Absorbed by U-238 Leak Out J.E. Crl.st
Resonances