When examining the radiation emitted from a specimen which has been irradiated in a reactor, considerable attention must be given to normalization and calibration of the data collected. When a specimen is irradiated the activity rises with increasing irradiation time (ti), and with increasing neutron flux level (or power). The sketch illustrates the time-based behavior of a typical single isotope during and following irradiation in a nuclear reactor.
The counts recorded by the detector can be expressed in terms of the parameters of the specific situation:
which becomes:
where:
h = the overall efficiency of the detector set-up, including effects of geometry, detector response, and self-shielding.
m = mass of the irradiated element, g
% = abundance of the isotope of interest
NA= Avagadro's number, atoms/g atom
sa= 2200 m/s microscopic absorption cross-section of the irradiated species, cm2
A = atomic mass of the irradiated element, AMU
f = thermal neutron flux, neutrons/cm2/s
f = branching ratio for the gamma-ray of interest
l = decay constant of the radioisotope produced, sec
1.128= correction for a Maxwellian distribution of neutrons
It is clear that in order to determine the mass of a given material in a sample using neutron activation analysis all parameters must be known or measured. Material constants are known and available. Careful planning and attention to detail will assure accurate measurement and recording of times. However, there are five parameters that present a problem:
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We can use the above equation, applied to two samples, irradiated in two separate runs, to derive the relationship between the masses of the target element in each. Assume the irradiation time and the count time for both samples are identical, and:
Then, by solving the equation for the target mass in sample one and sample two, then dividing one equation by the other, we arrive at the desired relationship:
Note that, for this well-controlled situation, the efficiency, abundance, and branching ratio cancel.
The remaining variable, the neutron flux seen by each sample, is difficult to manage with precision. The influence of variations in neutron flux from irradiation to irradiation is accounted for by use of a companion foil technique.
In this situation, every sample to be irradiated is accompanied by a foil. The sketch shows a typical polyethylene vial used for dry sample irradiation with a companion foil. A spare lid is used at the bottom of the vial to hold the foil in position during the irradiation. If shadowing of the sample by the foil is deemed a problem, a polyethylene standoff can be utilized to separate both components while maintaining the geometry from sample to sample.
Maintenance of the geometry, including position of each component in the core is essential. While gold foils are usually used because of their purity, reasonable half-life, and monoenergetic gamma emission, other foils may be used. Lets establish the subscript notation: 1 and 2 for the first and the second sample, and g1 and g2 for the first and the second companion foils. Using the previously presented formulation the counts recorded (G) when measuring any gold foil (g) irradiated in a neutron flux ff would be:
Note that % and f are both one for gold.
For the case of a gold foil having a half-life of 2.69 days, and for short irradiation times, and for short count times, the value of , and we arrive at values:
Further, if we utilize similar geometry foils, count using the same geometry, and counter settings (same set-up), identical irradiation and counting times, we find that:
Proceeding, if we take the neutron flux ratios for the foil and sample sets to be the same ( a reasonable presumption, given the similarity of sample/foil set-ups and use of the same irradiation position), we find that:
Note that by making the first measurement C1 a calibration exercise by "spiking" a sample similar in geometry and material to sample 2 with a known amount (mc1 )of the element of interest, the mass of sample 2 can be calculated from the measured data. The "spike" should be in the range of the mass of element anticipated in the sample(s) being analyzed.
Note that this technique requires knowledge of the elements contained in the sample being analyzed, and subsequent use of "spiked" sample(s) for each of those elements. In practice, a trial irradiation of an unknown sample, examined with a multi-channel analyzer will provide the information required to determine the series of calibration standards needed.
While the above process will produce excellent results, it is time-consuming and may require expensive and/or unavailable calibration elements. For many activities, the following broad-spectrum calibration of a multi-channel analyzer will provide adequate performance.
We begin with a calibration run(s), irradiating a representative sample(s) "spiked" with a number of pure elements providing j [ j = 1 to n] calibration peaks. Note that the gold companion foil technique (above) must be used to normalize the results to one of the runs if several samples are used. Thus, the counts in the jth peak (Cj), normalized to the rth run, will be:
We know, or can measure everything but hj and f. Thus: or:
where:
And, for the entire set of n gamma rays, we can plot [hjfr] as a function of the energy of the gamma ray, constructing a smooth curve through the n points.
Then, if we have a gamma ray at any energy Ex, , we can read [hfr]x from the curve. Finally, the mass of that element is related to the counts accumulated:
Note that we have not measured a sample "spiked" with a material that produces the gamma-ray of interest: we have deduced [hjfr]x from the series of measurements of other gammas. In addition, we must have included a companion gold foil (mgr) with the calibration run.
In practice, the measurement of the sample of interest yields Cx counts.
The following analysis assumes for all samples:
We know that the mass of the element of interest in the sample is given by:
Given the above constraints, [hjf] is a linear function of f, so that:
We have already shown how to employ gold foils to normalize measurements for flux variations:
Note that Fx is evaluated using the materials constants for the element/isotope of interest and the times associated with the actual sample irradiation, and includes the gold foil normalization term.
It should be evident from the sketch that analysis of the effect of source geometry is complex, so much so as to require use of sophisticated computational methods (such as Monte Carlo calculation) for solution.
However, we can conduct a "first order" estimate (it will probably under-predict the expected count) of the effect by assuming that all gammas travel vertically downward through the sample. For scintillation or solid-state detectors, we are interested in the photopeak, so the analysis looks only at the unscattered gamma component reaching the detector.
The gamma-ray source is considered to be uniformly distributed, with S gammas/s emitted downward. In this model, if no absorption occurred in the sample, S gammas/s would reach the detector and produce co counts/s (defining a detector efficiency ).
When the source is uniformly distributed throughout the sample, the effect of absorption in the sample is found by taking an element of volume at some height above the detector, y, and evaluating the number of uncollided gamma-rays that reach the detector:
The source strength at height y is .
If the density of the sample is r g/cm3, and has an attenuation coefficient m cm2/g, the number of gammas originating at dy and reaching the detector is
This results in a count rate contribution
Integrating over the height yields the count rate:
We see that the self-absorption factor becomes
Finally, we note that the self-shielding factor then depends on:
