Reactivity Coefficients
B. Rouben
McMaster University
EP 4P03/6P03
2016 Jan.-Apr.
2016 January
1
Reactivity Changes
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In studying kinetics, we have seen how
insertions of reactivity drive flux and power
changes.
Insertions of (positive or negative) reactivity
may come from:
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Refuelling operations (very slow change)
Saturating-fission-product (Xe, Sm, …) transients
(fairly slow change)
Sudden accidents or perturbations, which change one
or more lattice parameters (e.g., fuel, coolant, or
moderator temperature, coolant density, poison
concentration, etc…).
2016 January
2
Definition of Reactivity Coefficient
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A reactivity coefficient is defined as the derivative of the system
reactivity with respect to the change in a lattice parameter.
For instance, we can define:

Fuel  temperature T f  coefficient of reactivity 
T f

Moderator  density M d  coefficien t of reactivity 
M d
Coolant  density Cd  coefficient of

reactivity 
Cd

Power ( P ) coefficien t of reactivity 
, etc.
P
2016 January
3
Significance of Reactivity Coefficients
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It is important to know, for any given reactor design, the
sign and magnitude of the various reactivity
coefficients, as these coefficients suggest the
consequences of sudden changes in the operating
parameters:
A positive value for a reactivity coefficient means that a
positive change in that parameter will increase reactivity
and tend to increase power.
A negative value for a reactivity coefficient means that
a positive change in that parameter will decrease
reactivity and tend to decrease power
In both cases, a larger absolute value of the reactivity
coefficient  greater sensitivity to changes in that
parameter.
2016 January
4
Important Reactivity Coefficients
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Although they are not the only ones, the following
reactivity coefficients are particularly important:
Fuel-temperature reactivity coefficient, as fuel
temperatures will change in any power manoeuvres
Coolant-density reactivity coefficient, as coolant
density will change with the amount of boiling, and, in
safety analysis, coolant voiding (as a result of a Lossof-Coolant Accident) is extremely important to analyze
Power coefficient of reactivity, which combines effects
from the above two coefficients (and perhaps others)
2016 January
5
Units for Reactivity Coefficients
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Reactivity coefficients have the units of reactivity per
unit of the parameter against which the reactivity is
measured.
Since reactivity is a pure, unitless number, or can
alternatively be given in, say, mk, examples of units for
reactivity coefficients are:
-1
 mk/C degree (or degC ) – for a temp coefficient
3
 mk/(g/cm ) – for a density coefficient
 mk/%FP – for the power coefficient
2016 January
6
The Fuel-Temperature Reactivity Coefficient
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The fuel-temperature coefficient is governed by the
effect of temperature on the neutron absorption by fuel
Neutron absorption in fuel is marked by the existence of
resonances, in which neutron absorption is very high at
certain, very specific neutron energies (speeds) – see
sketch in the next slide.
If the neutron has a speed which exactly matches the
resonance energy, then there is a high probability of
absorption in a collision between the neutron at that
speed and the nuclide.
2016 January
7
This sketch is a log-log plot. Probability of absorption at a resonance
energy is orders of magnitude higher than at neighbouring energies. A
“resonance” region exists in the intermediate energy range [~1 ev -100
keV].
2016 January
8
Effect of Fuel Temperature
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Fuel temperature is the reflection of the random motion of fuel
nuclides – the higher the fuel temperature, the higher this random
“jiggling”.
Because of the jiggling, there is a range of relative speeds
between the neutron and the fuel nuclides, even for a fixed
neutron speed.
This means that, at higher fuel temperatures, neutrons with speed
slightly “off” the resonance energy can still be absorbed in the
resonance.
The effect is that the resonance is broadened at higher
temperature – this is called Doppler broadening – see next slide.
Even though the resonance peak is at the same time lowered
somewhat, the overall result is that there is more absorption in the
resonance at higher fuel temperature.
2016 January
9
Doppler Broadening of Resonance with Fuel Temperature
[from Nuclear Reactor Analysis, by James J. Duderstadt and Louis J. Hamilton, John Wiley & Sons, 1976]
2016 January
10
Effect on Reactivity Coefficient
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Because of the Doppler broadening of the absorption
resonances in fuel, the fuel-temperature reactivity
coefficient is negative:

 0. A typical value might be  0.01 mk / 0 C.
T f
Note: The fuel-temperature coefficient is a very prompt
effect – because fuel-temperature changes most quickly
in a change in power.
In an accident where the power increases, a negative
fuel-temperature reactivity coefficient provides a prompt
negative feedback, which tends to bring power back
down.
2016 January
11
Effect of Pu-239
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Although the fuel-temperature reactivity coefficient is negative,
the presence of Pu-239 in fuel makes it less negative.
This comes about because resonances are not always capture
resonances.
There are some fission resonances, in which increased absorption
means more fissions – therefore a positive reactivity effect!
Pu-239 has an important low-lying fission resonance at ~0.3 eV
neutron energy. This is very important because it is within the
thermal energy range, where the neutron flux is high.
As Pu-239 builds up with increased burnup, the fuel-temperature
reactivity coefficient becomes less negative!
The Pu-239 component is particularly important in CANDU
reactors, where the fuel is not enriched. Thus the fuel-temperature
reactivity coefficient in the equilibrium CANDU core may be
~-0.005 mk/oC.
2016 January
12
Pu-239: Low-Lying Fission Resonance at 0.293 eV
2016 January
13
Reactivity Insertion on Shutdown
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With a negative fuel-temperature reactivity coefficient,
the reduction in temperature when the reactor is shut
down will result in a positive reactivity insertion.
In CANDU, this reactivity insertion may be in the
range 5-10 mk, depending on the core burnup, and
depending on whether it’s a “hot” or “cold” shutdown
(i.e., to the coolant temperature of ~260 oC, or to room
temperature, 20 oC).
A reactivity device must be available to counter this
positive reactivity on shutdown, to ensure core remains
subcritical: e.g., the Mechanical Control Absorbers
(MCAs), or moderator poison.
2016 January
14
Coolant-Density Reactivity Coefficient
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In LWRs, where the coolant and the moderator are not separated,
a reduction in coolant density is equivalent to a reduction in
moderator density, which is a negative reactivity effect.
Turning this around, an increase in coolant density is a positive
reactivity effect in the LWRs:

i.e., in LWR,
0
Cd
2016 January
15
Coolant-Void Reactivity
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In reactor physics, one often speaks of the coolant-void reactivity (CVR). This
is not a coefficient, but rather it is the reactivity effect of losing all the coolant.
It is important to know this effect in the reactor safety analysis.
While it is not a coefficient (i.e., a derivative), we can see that the void
reactivity will generally have the opposite sign to the coolant density reactivity
coefficient (since it corresponds to a reduction – not an increase - in coolant
density).
Therefore, in LWRs, CVR < 0 (and large in absolute value).
However, in the standard CANDU, CVR >0.
This is because in the standard CANDU a loss of coolant is not equivalent to a
loss of moderator. There are subtle reactivity effects explained in the next
slides.
CVR > 0 does not make standard CANDU reactors unsafe! There are 2 fastacting, fully-capable, independent emergency shutdown systems, each of which
can mitigate the power excursion from a loss-of-coolant accident.
2016 January
16
Differential Effects on Voiding
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In a pressure-vessel reactor, the coolant and moderator are not
separated. Here coolant voiding is equivalent to loss of moderator,
 large negative reactivity, reactor shuts down.
But CANDU is a pressure-tube reactor  the loss of coolant is not
a loss of moderator.
In fact, in the standard CANDU, the coolant contributes little to
moderation, and coolant loss gives a positive void reactivity.
We will consider how neutron events are changed when coolant is
lost, and the effect on reactivity.
2016 January
17
CANDU BASIC-LATTICE CELL WITH 37-ELEMENT FUEL
Face View
of a Bundle
in a Fuel
Channel
D2O
Primary
Coolant
Gas Annulus
Fuel Elements
Pressure Tube
Calandria Tube
2016 January
Moderator
18
Standard-CANDU Coolant-Void Reactivity
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Before Neutrons Leave a Channel
Before escaping from the channel where they are born,
some fission neutrons are normally slowed by coolant
into the resonance energy region and are absorbed.
Now imagine the coolant is lost. Without coolant, the
following will happen:
 Fewer fast neutrons will be slowed into the resonance
region, therefore there will be more opportunities for
fast neutrons to induce fission (more fast-fission
production):  > 0, and
 More fast neutrons will escape resonance capture and
reach the moderator (less absorption): p > 0
 Both phenomena increase reactivity
cont’d
2016 January
19
Standard-CANDU Coolant-Void Reactivity
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After Neutrons Re-enter a Channel
Upon entering a channel from the moderator, some
thermalized neutrons are scattered by hot coolant to
higher energies and resonance capture.
Now imagine the coolant is lost. Without coolant,
scattering to higher energies does not occur, and more
neutrons escape resonance capture.
 This gives rise to a positive reactivity change from U238: pU-235 > 0, but
 To a negative reactivity change from Pu-239 (on
account of the resonance at 0.3 eV): pPu-239 < 0
cont’d
2016 January
20
Standard-CANDU Coolant-Void Reactivity
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The overall result of the 3 positive components and the 1 negative
component is that the net CVR of the standard CANDU is
positive, but decreases as Pu-239 builds up with burnup:
CVR (initial core – all fuel fresh)  +20 mk
CVR (equilibrium core – mixture of burnups) +15 mk
Note that it is not physically possible to lose all the coolant
instantaneously – therefore there cannot be an instantaneous
insertion of +15 mk.
In addition, reactivity insertion in a large LOCA can be reduced by
subdividing the coolant into more than one loop (and having
bidirectional flow) - see next slide.
Therefore, the reactivity insertion in a large LOCA may be of the
order of 4-5 mk in the first second after the break.
Each shutdown system can be actuated within 1 s, and can insert a
large negative reactivity (e.g., -50 mk) in the first second after
actuation.
2016 January
21
Non-Uniform Voiding Transient
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Coolant voiding in a large
LOCA is not uniform.

Side-by-Side Heat-Transport-System Loops in
CANDU 6
For instance, in the
CANDU 6, the heat
transport system is
subdivided into two
side-by-side loops,
each servicing one
half of the
cylindrical reactor.
2016 January
22
Reducing Coolant-Void Reactivity
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Even though a positive CVR may not mean poor safety, there may
be a negative perception of a positive CVR. Then how would one
go about reducing the CVR to deal with this perception?
One way: Give coolant greater role in moderation. Increase ratio
of coolant volume to moderator volume, e.g. by reducing lattice
pitch, and/or increasing outer diameter of pressure tube. Would
reduce reactivity even in cooled state  need fuel enrichment.
Another way: Make use of flux redistribution on coolant voiding
(relative flux increase in centre of bundle). Inserting poison
material in central pin will increase absorption on coolant loss 
reduced void reactivity.
Poison in central pin used in Low-Void-Reactivity Fuel
All these options used for ACR-1000 (see comparison of basic
cells in next Figure).
2016 January
23
Lattice-Cell Comparison
CANDU-6
Cells
ACR Cells
2016 January
24
Power Coefficient of Reactivity
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An increase in power results in a prompt increase
in fuel temperature, and may result in an increase
in coolant boiling.
Thus, the power coefficient of reactivity will in
general be a combination of an increase in fuel
temperature ( < 0) and a reduction in coolant
density ( < 0 in LWR, >0 in standard CANDU)
2016 January
25
Moderator-Poison Reactivity Coefficient

Since a moderator poison simply absorbs neutrons, we
can see immediately that

0
Pc
In fact in CANDU :
Boron reactivity coefficient  8 mk / ppm B
Gadolinium reactivity coefficien t  28 mk / ppm Gd
2016 January
26
END
2016 January
27