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 . 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 .
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:
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-2s-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 :
where , , ,
, , – actual physical dimensions of the core,
– extrapolation length.
Accordingly, the distribution of neutron flux along axis z at a given place x0, y0 can be written:
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:
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
where – thermal neutron flux [m-2s-1],
– microscopic activation cross section [m2],
– 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:
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):
The activity of the detector at the end of irradiation enduring time T :
The activity at time t after finishing the irradiation:
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:
Finally, the time function of the detector activity is obtained by using the above formulae:
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:
where k – Boltzmann constant,
T – neutron temperature [K],
m – mass of neutron [g].
At room temperature (20 °C = 293 K) v0 = 2200 m/s which corresponds to an energy of 0.025 eV. Again, according to the Maxwell-Boltzmann distribution, the average velocity of neutrons:
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:
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 v0 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:
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 Tn is given by:
and T0 = 293 K.
The number of target nuclei NT :
where m – mass of sample [g],
a – abundance of target isotope,
A – mass number of target isotope [g],
L - 6.02 × 1023.
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:
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:
On the basis of (19) and (20), the ratio of activity induced by thermal neutrons to the total activity of the detector:
In case of detector materials with activation cross section following the 1/v law through the neutron energy range, the value of RCd 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 197Au 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:
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 198Au by neutron capture can be neglected because of the experimental conditions (Fth, sact, 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):
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:
where h – efficiency of counting apparatus.
The measurement delivers the total number of counts during a finite measuring period tm:
The activity of the sample:
If the measuring period is short (l tm » 0), then B » h A (t) tm , and A(t) » B / (h tm).
The number of counts during the measurement can be determined using formulas (23) and (27):
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) ~ Fth(z)) if the value of function
is constant k. This will be fulfilled if the measuring period and the time of measurement are related by:
This relationship controls the choice of period tm for a measurement starting at a time t after irradiation.
The method used in the exercise to ensure adequate measuring periods tm 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 Bw coming from the scanned part of the wire, and the timer (chain I), taking the counts BI 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
· 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 28Al 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 165Dy 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 (90Sr-90Y, T1/2 = 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 Bi = B(zi) 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 Bi 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:
(n being the number of measured points).
There are 3 unknown values such as B0, z0, C in formula (32). The calculations can be considerably simplified by predetermining B0 as the smoothed (averaged) value of the maximum counts at the middle of the wire.
The remaining unknowns, z0 and C are convenient to determine via a suitable transformation. By rearranging relationship (32):
the problem reduces to a linear regression problem:
With the well known relationships of linear regression:
yielding the two parameters in question:
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):
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:
Similarly, the effect of “inaccurate” time setting on the wire counts Bw :
From the latter three relationships:
If the measuring period is short (l tm » 0), the ratio in (44) will approximately agree with the ration of the counts:
The standard deviation of the counts of the wire shall be added to this standard deviation:
The resultant standard deviation is:
For rating the measurement, the relative standard deviation of interest is:
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):
where Ib , ICd – 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?
Duderstadt, J. J., Hamilton, L. J.: Nuclear Reactor Analysis, John Wiley
New York, 1976.
 Bell, G., Glastone, S.: Nuclear Reactor Theory, Litton Educational Publishing, Inc., 1970.
 Glasstone, S., Edlund, M.C.: The Elements of Nuclear Reactor Theory, Van Nostrand, New York, 1956.