**CONTENT**

DETERMINATION OF THERMAL NEUTRON FLUX

Neutron spectrum, neutron flux distribution in thermal reactors

Determination of the thermal neutron flux by activation method

Determination of the thermal neutron flux from the activity of the
irradiated detector

Determination of the absolute value of thermal neutron flux by means
of a gold foil detector 5

Determination of the relative distribution of thermal neutron flux

Principle of measurement of the wire activity; description of the
measuring apparatus

Equipment and Materials Needed for the Measurement

Measurement of the relative distribution of the thermal neutron flux

Measurement to determine the
absolute value of thermal neutron flux

Data Evaluation - Uncertainty Estimation

Evaluation of relative distribution of thermal
neutron flux

Determination of the parameters of the cosine curve

Determination of the absolute value of thermal neutron flux

** **

The nuclei of certain naturally occurring isotopes
can be transformed into radioactive ones by exposing the materiel to neutron
radiation, and the activity of the radioactive products produced can be
measured by means of appropriate counter system. In addition to the factors
determined by the conditions of measurement, this activity is affected only by
the neutron flux in the point of irradiation and by the activation cross
section of the target material which in the present case, is the neutron
detector. Provided the activation cross section is known, the neutron flux can
be determined by measurement of the activity of the sample irradiated. As the
activation cross section of a number of materials depends on the neutron energy
in different way this method will yield information also on the energy
distribution of neutrons.

In the present measurement the thermal neutron flux
will be determined using activation detectors.

In the nuclear reactors, practically only fast
neutrons are produced in the fission reactions. Energy distribution of prompt
fission neutrons is according to the Watt’s spectrum [1]. In thermal nuclear
rectors, the fast neutrons gradually slow down due first of all to the
collision with the atoms of the moderator and, as a result of the various
nuclear processes taking place in the reactor, the so-called reactor spectrum
will develop [1].

Of the reactor spectrum, thermal neutrons shall be
treated distinctly, these being decisive for the behavior of thermal nuclear
reactors. These neutrons are in equilibrium with atoms of the medium, their
velocity distribution being approximately described by the Maxwell-Boltzmann
distribution.

Let *v*
designate the characteristic velocity of thermal neutrons, the thermal neutron
flux is defined as follows:

** _{} , ** (1)

where _{} is thermal neutron
density around a point of vector r.

The thermal neutron flux defined this way is a
scalar value of dimension m^{-2}s^{-1}, and it expresses the
total path of thermal neutrons in unit volume around point r in a second.

In thermal reactors, the
spatial change of neutron flux is significant. In case of cuboid-shaped uniform
bare (without reflector) reactor core, the spatial distribution of thermal
neutron flux, according to the reactor theory, is given by the following
relationship [1]:

** _{} , **(2)

where _{}, _{}, _{},

_{}, _{}, _{} – actual physical
dimensions of the core,

_{} –
extrapolation length.

Accordingly, the distribution of neutron flux along
axis *z* at a given place *x*_{0}, *y*_{0} can be written:

_{} , (3)

where

_{} .

The non-uniform neutron flux distribution is
unfavorable, among others in that it results in a non-uniform burn-up of the
fuel rods, the central rods burning out soon due to the high flux density while
rods arranged in the edge of the core are hardly burned up. This problem may be
eased by using a reflector surrounding the reactor by a material scattering a
considerable part of neutrons leaking through the reactor boundary back into
the core. As a result, the thermal neutron flux increases with explicit
reflector peaks in the thermal neutron flux distribution on the edge of the
reactor (Fig. 1.)

Figure 1. Thermal neutron
flux distribution in a reflected reactor

Use of a reflector improves economy of the reactor
construction, as less fuel is required to achieve the critical mass. The
so-called reflector saving means the decrease of the linear dimensions of the
critical reactor due to the reflector. Neglecting the extrapolation length it
can be determined at a good approximation by fitting a function (in the present
case a cosine function) to the neutron flux distribution developing in the core
of the reactor and extrapolating the curve to the reflector area. The
extrapolated boundary of the bare critical reactor equivalent to the reflected
reactor i.e. which would produce the same neutron flux distribution within the
core as does the reflected reactor would be at the zero value of this curve
(i.e. at zero flux (Fig.1.)).

Hence, reflector saving is understood as the
difference between the linear dimensions of the equivalent bare critical
reactor and the reflected critical reactor:

_{}, (4)

where *C* – distance
between the zero value points of the cosine curve fitted to the neutron flux
distribution inside the core,

_{} – actual physical dimension of the core.

Irradiating
an activation detector in thermal neutron flux the reaction rate during
activation is expressed by

_{}, (5)

where _{} – thermal neutron flux [m^{-2}s^{-1}],

_{} – microscopic activation cross section [m^{2}],

_{} – number of target atoms in the sample.

The rate of change of radioactive atoms during
irradiation is the difference between the rates of production and decay:

_{}, (6)

where *l** *– decay
constant of the radioactive nuclei produced [s^{-1}],

*t* –
time [s].

Be _{} at the start of irradiation.
Then the solution of the differential equation (6) using (5):

_{}. (7)

The activity of the detector at the end of
irradiation enduring time *T* :

_{}. (8)

The activity at time *t* after finishing the
irradiation:

_{}. (9)

These relationships will,
however, be accurate only if each atom of the sample is irradiated by the same
neutron flux. In case of a foil of non-negligible thickness, the average
neutron flux will be lower inside it (_{}) than on the surface (_{}), due to neutron absorption. Their ratio is the
self-shielding factor:

_{}. (10)

Finally, the time function of the detector activity
is obtained by using the above formulae:

_{}. (11)

In the thermal neutron energy range, the velocity
distribution of neutrons follows the Maxwell-Boltzmann distribution according
to which the most probable velocity of the neutrons is:

_{}, (12)

where *k*
– Boltzmann constant,

*T* – neutron
temperature [K],

*m* – mass of
neutron [g].

At room temperature (20 °C = 293 K) *v*_{0}
= 2200 m/s which corresponds to an energy of 0.025 eV. Again, according to
the Maxwell-Boltzmann distribution, the average velocity of neutrons:

_{}. (13)

The reaction rate defined
by (5) describes the activation process for irradiation in a monoenergetic
neutron flux. As the thermal neutron velocities are different and the cross
sections are dependent on the neutron velocity *v*, the activation process would be correctly describes by:

_{}. (14)

Generally, this would be seriously complicate
procedure and, therefore, the integral above is substituted by the product in
(5) where the values _{}and _{}will be determined in concordance with each other in such a
way that their product yields the actual reaction rate according to (14). For
instance, it is customary to define neutron flux _{}, where *v*_{0}
is the most probable velocity of neutrons according to (12), *n* is the total density of thermal
neutrons. In this case, the reaction rate for a 1/*v *detector (*s* ~ 1/*v*) and non-distorted Maxwell-Boltzmann spectrum will be obtained
from:

_{}, (15)

where _{} – activation cross section belonging to the
most probable velocity of Maxwell-Boltzmann distribution.

Defining the thermal neutron flux as _{} (where _{} is the average
velocity of thermal neutrons according to (13)), involves a cross section _{}which, in case of a cross section ‘1/*v*’ and a neutron temperature *T*_{n}
is given by:

_{} (16)

and
*T*_{0} = 293 K.

The
number of target nuclei *N*_{T}
:

_{} (17)

where *m* –
mass of sample [g],

*a** * – abundance
of target isotope,

*A * –
mass number of target isotope [g],

L - 6.02 × 10^{23}.

From the detector activity
in (11), only the activity induced by thermal neutrons shell be taken into
account.

Irradiating the detector
at a place where also a significant epithermal flux is present and/or the
epithermal activation cross section of the detector is significant compared to
the thermal activation cross section, then a part of the activity obtained is
due to the epithermal neutrons. Therefore a correction has to be applied to
determine the contribution of epithermal neutron flux to the activity produced.
This is possible because the neutron absorption cross section of cadmium is
very high at thermal energies (Fig. 2.), while negligible in the epithermal
range. Hence, if the detector covered with cadmium of appropriate thickness,
then thermal neutrons will be practically fully absorbed by the cadmium so that
the activity of reactor will be induced only by epithermal neutrons of energies
above about 0.4 to 0.7 eV, depending on thickness of the cadmium cover.
Activity induced by thermal neutrons is simple to determine by irradiating two
uniform detectors, one bare and the other covered with cadmium:

_{}, (19)

where _{} – activity of bare detector,

_{} – activity of cadmium-covered detector.

It is also customary to define the so-called
cadmium ratio, the ratio of the activities of bare and cadmium-covered detectors:

_{}. (20)

On the basis of (19) and (20), the ratio of activity
induced by thermal neutrons to the total activity of the detector:

_{}. (21)

In case of detector materials with activation cross
section following the 1/*v* law through
the neutron energy range, the value of *R*_{Cd}
is very high an thus, according to (21), the share of epithermal neutrons in
the activity induced in the detector may be neglected.

In the range of thermal neutron energies, the
activation cross section of ^{197}Au follows the law 1/*v* but it has a number of resonances.

Figure 2. Total
microscopic cross section of Dy, Au and Cd vs. neutron energy

Using a bare detector and
a cadmium covered detector of identical masses, the thermal neutron flux is
calculated from relationships (11), (16), (17) and (19) as follows:

_{}. (22)

The following series of reactions will take place in
gold irradiated by neutrons:

Of this multi-step reaction series, actually it is
sufficient to take the first one into consideration because the activation of ^{198}Au
by neutron capture can be neglected because of the experimental conditions (*F*_{th}, *s*_{act}, irradiation time etc.,
see later).

ce peaks in the resonance region (see Fig. 2.) stressing the role of cadmium correction according to (19).

The following reaction is applied to determine the
relative spatial distribution of thermal neutron flux:

The activation cross section of dysprosium is very
large and approximately 1/*v*-dependent
throughout the neutron energy range. Accordingly, the contribution of
epithermal neutrons to its activity is negligible. Hence, the activity of such
detectors is proportional to the thermal neutron flux and, therefore, the
counts during the measurement can be directly used to calculate the relative
thermal neutron flux distribution. The simplest way to provide identical
experimental conditions (identical volume of samples, identical irradiation
time etc.) is to use a wire sample to be activated and measuring the activity
of wire sections. However, the activity of the sample decreases continuously in
the course of measurement, which has to be compensated during the measurement
or correction has to be applied for decay in evaluating the results. In this
exercise, we prefer compensation.

The activity of the detector at the start of the
measurement can be written on the basis of formula (11):

_{}. (23)

Provided the activity of the detector is not
reasonably large compared with the dead time of the counting apparatus, the
indicated intensity *I*(*t*) will be proportional to
the sample activity:

_{} (24)

where *h* – efficiency of counting
apparatus.

The measurement delivers the total number of counts
during a finite measuring period *t*_{m}:

_{} (25)

or

_{} (26)

Hence:

_{}. (27)

The activity of the sample:

_{}. (28)

If the measuring period is short (*l** t*_{m} » 0), then *B* » *h** A *(*t*) *t*_{m }, and *A*(*t*) » *B / *(*h** t*_{m}).

The number of counts during the measurement can be
determined using formulas (23) and (27):

_{}. (29)

The subproduct in figure brackets is the same for
each point of the wire. The number of counts will be proportional to the
thermal neutron flux (*B*(*z*,*t*) ~ *F*_{th}(*z*)) if the value of function

_{} (30)

is constant *k*.
This will be fulfilled if the measuring period and the time of measurement are
related by:

_{}. (31)

This relationship controls the choice of period *t*_{m} for a measurement starting
at a time *t** *after irradiation.

The method used in the exercise to ensure adequate
measuring periods *t*_{m} will
be described below.

A wire of Dy-Al alloy is used to determine the
spatial distribution of the thermal neutron flux. The activity as a function of
the position along the Dy-Al wire is determined by a so-called wire activity
measuring apparatus. The radioactive decay during the measurement is
automatically corrected by the apparatus by increasing the time of measurement
of the subsequent wire sections in accordance with the extent of decay. To set
the appropriate measuring periods is guaranteed by a foil of identical
composition, and half-life as those of the tested wire.

The scheme of the measuring apparatus is seen in
Fig. 3. Its principle of operation is as follows: the apparatus contains two
measuring chains, one to measure the activity of the wire (chain *E*) by recording the counts *B*_{w} coming from the scanned
part of the wire, and the timer (chain *I*),
taking the counts *B*_{I}
proportional to the activity of a foil made of the same material as, and
activated simultaneously with, the wire. This chain contains a preset counter
stopping the counter of chain *E* after
the preset counts have been reached, and simultaneously giving a signal to the
apparatus to forward the wire by the preset length. The length of steps can be
adjusted in the range of 1 to 10 mm on the apparatus. The measurement itself is
controlled by a dedicated computer program running on a PC.

Figure 3. Scheme of the
apparatus for measuring the activity of wire

· Reactor + pneumatic rabbit
system for sample delivery

· Plexiglass rod to hold the
wire

· Scintillation detector and
pulse counter apparatus

· Wire activity measuring
apparatus (controlled by a PC)

· Wire of Dy-Al alloy

· Foil of Dy-Al alloy

· Gold foils

· Al and Cd capsules for
covering the foils

· Long life standard
radiation source

· Stopwatch

· Radiation protection
equipment (rubber gloves, pincers etc.)

In this exercise, the relative distribution of the
thermal neutron flux will be determined along a vertical axis of the core,
using the method described in the chapter 2.2.4. Then, the vertical reflector
savings are determined. A wire of Dy-Al alloy containing 10% of Dy is used for
the measurement of the neutron flux distribution. The absolute value of thermal
neutron flux is determined by a gold activation detector.

In preparing the measurement, the wire is placed in
a plexiglass rod. (The moderation properties of plexiglass are similar to those
of water and thus the disturbance of the flux distribution in the core will be
negligible.)

The plexiglass rod is then placed by the operator to
the core position E6 of the non-operating reactor and the irradiation is
performed. Simultaneously also the dysprosium foil is activated in the rabbit
system in the core position D5. Since the wire used is made of Dy-Al alloy, the
samples have to rest for at least 20 minutes (half-live of the ^{28}Al
is 2.24 min) after irradiation to permit the interfering activity of Al to
decay. Then the wire is removed from the plexiglass rod and placed into the
wire activity measuring apparatus. The foil is placed below the detector of the
timer chain. Taking relationship (49) (see later) into consideration, _{}is preset in the controlling program and the measurement
started. A data sheet is supplied, containing the data required for setting the
measuring apparatus. In the measurement, *b**¯* particles from the decay
of ^{165}Dy are detected.

In the irradiation channel in core position D5 two
gold foils of identical size and mass – one bare while the other in cadmium
cover – are irradiated. The foils in polyethylene sample holder are forwarded
to the operating reactor by means of the pneumatic rabbit system joining the
irradiation channel. Record time of the end of irradiation and, using a
stopwatch, measure the period elapsed before the start of the measurements.
Sample intensity is taken by a scintillation counter. Because of *b**¯*
decay in activated Au foils, their activity shall be measured by means of
scintillation counter equipped with plastic phosphorus. A data sheet is
available for the measurement, containing the *b**¯ *counting efficiency of the
apparatus and the data required for setting its operating parameters. Before
measurement make sure that setting of the device is correct. For this purpose
take the counts of a long half-life standard radiation source (^{90}Sr-^{90}Y,
*T*_{1/2} = 28.5 a) and compare
them with the reference value. Apply correction if necessary. Measure also the
activity of the background and correct for it in evaluating the data measured.

Applying the correction principle described in the
chapter 2.2.4. the spatial distribution of the counts *B*_{i} = *B*(*z*_{i}) will also give the
thermal neutron flux distribution. Plotting the data obtained and approximating
the neutron flux distribution in the core by a cosine function, determine the
reflector savings in accordance with the chapter 2.1.

In the measurement of wire activity a length of step
*S* is used. The counts *B*_{i} are assigned to points *i·S *(*i*
= 1, 2, 3, …). As the argument of the function *j* (see eq. (3)) is the
co-ordinate *z*, also the different
points of the wire and co-ordinates *z*
have to be matched. The aim is to minimize the sum of squared differences:

_{} (32)

(*n* being
the number of measured points).

There are 3 unknown values such as *B*_{0},* z*_{0},* C* in
formula (32). The calculations can be considerably simplified by predetermining
*B*_{0} as the smoothed
(averaged) value of the maximum counts at the middle of the wire.

The remaining unknowns, *z*_{0} and *C* are
convenient to determine via a suitable transformation. By rearranging
relationship (32):

_{} (33)

the problem reduces to a linear regression problem:

_{}

_{}

_{} (34)

_{}

_{}

With the well known relationships of linear
regression:

_{} (35)

_{} (36)

yielding the two parameters in question:

_{} (37)

_{} . (38)

The control program of the wire measuring apparatus
offers an option to perform the fitting procedure described above.

Choosing the measuring condition so that the
background counts are negligible, the standard deviation of counts is
determined by uncertainty propagation analysis.

The uncertainty in setting the measuring time is
obtained from (27):

_{} , (39)

_{} . (40)

During the measurement of a radioactive isotope the
distribution of the counts measured in uniform time-intervals follows the
Poissonian distribution. Therefore, the standard deviation of the measured
counts is equal to the square root of the counts (provided no other factors are
acting). If the preset value of the timer chain is _{}, then the standard deviation in timing will be:

_{} . (41)

Similarly, the effect of “inaccurate” time setting
on the wire counts *B*_{w} :

_{} (42)

_{}. (43)

From the latter three relationships:

_{} . (44)

If the measuring period is short (*l** t*_{m} » 0), the ratio in (44) will approximately agree with
the ration of the counts:

_{}. (45)

The standard deviation of the counts of the wire
shall be added to this standard deviation:

_{}. (46)

The resultant standard deviation is:

_{} , (47)

_{} . (48)

For rating the measurement, the relative standard
deviation of interest is:

_{} . (49)

The accuracy in fitting the curve will be
satisfactory if the deviation of the different measured points from the
“smooth” function is below about 1% on the average, requiring to keep the
relative standard deviation according to (49) below about 0.3 to 0.4%. This
will be taken into consideration in selecting the preset value of counts and
the counts (activity) of the wire.

Knowing the activity of the gold foil detector, the
thermal neutron flux is determined from relationship (22). A data sheet will be
available for the measurement containing the values of nuclear data and mass
data required for the calculation. The relative uncertainty of the result
obtained, determined by the conditions of measurement, can be determined from
relationship (22) (assuming Poisson’s distribution):

_{} , (50)

where *I*_{b}
, *I*_{Cd} – activity
of bare and cadmium covered samples, respectively, without correcting for the
activity of the background (*H*),

*h* –
*b**¯ *counting efficiency of the
counter (see(24)).

1.
How do we define the scalar neutron flux? What is its dimension?

2.
Characterize the spatial distribution of thermal neutron flux in a bare
and a reflected reactor.

3.
What is the principle of the determination of thermal neutron flux by
activation method?

4.
Why do we use cadmium cover for one of the gold samples and how do we
define the cadmium ratio?

5.
Why is it not necessary to use cadmium cover for the dysprosium (Dy-Al alloy)
wire?

6.
What is the principle of decay compensation applied for the measurement
of the activity of Dy-Al wire?

[1]
Duderstadt, J. J., Hamilton, L. J.: Nuclear Reactor Analysis, John Wiley
& Sons,

New York, 1976.

[2]
Bell, G., Glastone, S.: Nuclear Reactor Theory, Litton Educational
Publishing, Inc., 1970.

[3]
Glasstone, S., Edlund, M.C.: The Elements of Nuclear Reactor Theory, Van
Nostrand, New York, 1956.