Contents lists available atScienceDirect
Applied Radiation and Isotopes
journal homepage:www.elsevier.com/locate/apradiso
The Lambert-Beer law in time domain form and its application
Volodymyr Mosorov
Institute of Applied Computer Science, Lodz University of Technology, Lodz 90-924, Poland
H I G H L I G H T S
•
The Lambert-Beer law in time domain form was proposed.•
Count rate measured by radiation detector was changed to registration of time intervals.•
Proposed approach requires the development of a new electronic part for a radiation detector.A R T I C L E I N F O
Keywords:
Radioisotope gauge Temporal resolution Time interval registration
A B S T R A C T
The majority of current radioisotope gauges utilize measurements of intensity for a chosen sampling time in- terval using a detector. Such an approach has several disadvantages: temporal resolution of the gauge isfixed and the accuracy of the measurements is not the same for different count rate. The solution can be the use of a stronger radioactive source, but it will be conflicted with ALARA (As Low As Reasonably Achievable) principle.
Therefore, the article presents an alternative approach which is based on modified Lambert-Beer law. The basis of the approach is the registration of time intervals instead of the registration of counts. It allows to increase the temporal resolution of a gauge without the necessity of using a stronger radioactive source and the accuracy of the measurements will not depend on count rate.
1. Introduction
The functionality required for radioisotope gauges in most cases is a question of measuring the remaining radiation intensity i.e., the amount of the energy which remains in the beam after passing through the examined area/volume in a given unit of time. Such intensity is measured by the radiation detector which is the core of the radioisotope gauge. In case of X-ray and gamma-ray radiation, the measurement of the remaining intensity is performed by sampling the number of pho- tons absorbed in the detector crystal in some particular time period.
This means that the detector measures the intensity for the chosen sampling time interval. The principle of detector operation can be found for instance inJohansen and Jackson (2004).
One of the fundamental parameters of a radioisotope gauge is its resolution (sensitivity). Similarly to Roy et al. (2002), the resolution can be defined as fractional change in photon counts recorded by the detector with a differential change in the inspected volume of interest.
One of the special kinds of resolution is temporal resolution. Temporal resolution of a radioisotope gauge can be defined as the shortest time interval to which the gauge will respond, with the change in the in- spected volume being registered or detected. Temporal resolution is related to the precision at which the measurement was made and to the minimum detectable change.
The probabilistic nature of radiation and thefinite value of the time observation cause that objectively the counts are random variables. For
longer time observations, the countNxcan be also treated as a mean of the number of the photons absorbed by a detector. In practice, the precision of the approximation ofNxcan be defined by standard de- viationσx, which will depend on a given time samplingTand the source strength. It is important to note that for a large number of counts, the standard deviationσx may be estimated as the square root of a large number of countsσx= Nx(Johansen and Jackson, 2004). However, in a case of small number of counts, count raterelativeerror (%) increases and, as a result, the precision of the measurements is lower. To con- clude, the precision of the measurements is not the same for different photon count rates. It is one of the disadvantages of current radio- isotope gauges. On the other hand, temporal resolution of a radio- isotope gauge based on counting photons absorbed in detector crystal is fixed and it is equal to a chosen time samplingT.
There are two possibilities to decrease the uncertainty of the counts and thereby increase the precision of the radioisotope gauge: a stronger radioactive source, i.e., a larger number of photons emitted per time unit by the source, and/or a longer time interval of data acquisitionT.
However, a longer intervalTmeans worse temporal resolution of the gauge, and less possibility of investigating more rapid processes. On other hand, selecting a stronger radiation source for an application is not simply a technical question. It must also comply with the ALARA (As Low As Reasonably Achievable) principle (Croft, 1991), which re- quires the minimization of radiation doses and releasing of radioactive materials. More than merely best practice, ALARA is predicated on legal http://dx.doi.org/10.1016/j.apradiso.2017.06.039
Received 26 January 2017; Received in revised form 20 June 2017; Accepted 25 June 2017 Available online 27 June 2017
0969-8043/ © 2017 Elsevier Ltd. All rights reserved.
MARK
dose limits for regulatory compliance and is a requirement for all ra- diation safety programmes.
To solve these contradictory requirements, we propose to apply the modified Lambert-Beer law where the measurement of intensity as countsNxwill be replaced by registration of time intervalsTx. The idea is to choose an arbitraryfixed number of countsNfand measure the time intervalTx, which is required for registration of the photonsNfby a detector. Section 2introduces the modified Lambert-Beer law and shows its advantages as compared with the current form. In the fol- lowing section, the proposed approach was verified via simulations for several applications. The limitations and drawbacks of the proposed approach are discussed in the conclusion.
2. The Lambert-Beer law in time form
The radiation intensity is the amount of energy passing through a volume of interest in a given unit of time and it can easily be measured with a proper detector. The attenuation of the beam of X-rays or gamma-rays in a material is described by the Lambert-Beer's law which in a general form may be written as:
= ⋅− ⋅
I I e0 μ l (1)
whereI0is the initial intensity of the radioactive source, I is the re- maining beam intensity,µis the linear attenuation coefficient andlis the thickness of the absorber.
In physical terms, the intensity of the beamIis the average power transferred over one period of the wave. In case of radioactive sources, the intensity is the number of photons per time unit (usually 1 s) through a given cross-section and, therefore, the Lambert-Beer law can be rewritten as
Fig. 1.Length of registered time intervals vs. thickness of absorbent for two cases of linear attenuation coefficients µ1, µ2(µ1< µ2).
Fig. 2.(a) Measured counts by a detector forfixed sampling timeT =1 ms (count rate relative errorσ, (%)of measured counts is changing) (b) series of suitable pulses gener- ated by hypothetical detector forNf= 100.
Fig. 3.Cross section of simulated facility for pulse-velocity measurement of aflow.
Fig. 4.a) Simulated counts measured by detector forfixed time samplingT=1 ms.
Maximum value means that particle is positioned on the detector plane level b) series of suitable pulses forNf=50.
Fig. 5.Cross section of simulated facility for two component traction measurement.
= ⋅
⋅− ⋅ N N T
T e
x 0 x μ l
0 (2)
whereN0is the initial number of photons emitted by the radioactive source during the time interval T0, Nx is the remaining number of photons,Txis a given time measurement interval.
Assuming thatNxis afixed value i.e.,Nx=Nf, whereNfwas a value chosen before arbitrarily, Eq.(2)can be rewritten in a different manner:
= ⋅
⋅ ⋅ T N T
N e
x
f 0 μ l
0 (3)
Eq.(3)can be named as the Lambert-Beer law in time domain form.
It is oblivious that the registered time intervalTxis depended on both
the material absorption propertiesµand the thickness of the absorbent l, i.e., the smaller the linear attenuation coefficient and/or thickness of the absorber is, the shorter the time interval will be.Fig. 1shows the dependence between the length of the registered time intervalTxand the absorber thicknessl. As observed inFig. 1, the length of intervals will grow exponentially when the thickness (or/and the linear at- tenuation coefficient) increases. However, for small values oflone can assume that time intervalTxwill increase linearly.
The registration of time intervals instead of the registration of counts will require the development of a new electronic part for a de- tector. However, these modifications will be easy enough and they re- quire the implementation of minor changes in the current types of de- tectors. Thus, such an upgraded detector will produce short pulses on the output and the time intervals between two neighboring pulses will correspond toTx. In computer unit these intervalsTxwill be converted into digits.
The choice of valueNfcan be made basing on the calibration pro- cedure of the radioisotope gauge. Usually, the minimum and the max- imum values of counts measured by a detector for a given gauge and measuring conditions are known; therefore, Nfcan be chosen as the minimum value of counts measured.
Fig. 2illustrates the proposed approach based on Eq.(3). The ex- ample represents the registration of the movement of a radioactive particle to the detector plane and away from it. The counts measured by a detector for thefixed sampling timeT =1 ms are shown inFig. 2a.
The series of appropriate registered pulses generated by a hypothetical detector (Nf= 100) is presented inFig. 2b. The shorter intervals mean the particle is close to the detector's plane level. As observed inFig. 2b, one can register rapid changes of radiation intensity contrary to the classical way. Thus, in this case, the registration of intervalsTxallows to determine the time when a particle will pass the detector's plane with temporal resolution approximately 0.2 ms (seeFig. 2b) instead offixed timeT =1 ms (seeFig. 2a).
Also,relativedeviationδ(%) of measurements is still constant con- trary to the classical approach when it is changing as noted above. The reason is that the relative standard deviationδis estimated as inversely proportional to the square root of thefixed number of countsNfwhich isfixed:
= = =
δ σ
N N
1 const
f
f f (4)
3. Simulation results
The use of the modified Lambert-Beer's law was confirmed by the results of simulations carried out for two classical applications of Fig. 6.Count rate vs. liquid volume fraction.
Fig. 7.Changes of liquid volume fraction in time.
Fig. 8.a) Simulated counts measured by detector forfixed time samplingT=300 ms b) series of suitable pulses forNf=850.
radioisotope gauges. All these simulations were performed in the MCNP5 environment (Monte Carlo Code Group, 2016), which gives excellent possibilities to simulate different types of radioisotope gauges.
Thus, the MCNP5 allows:
a. to define the 3D geometry of the radioisotope gauge, including the percentage and the density of all the materials
b. to model the radioactive source
c. to simulate the detector type, and to record the energy distribution of pulses created in a detector.
The examples of the use of the MCNP5 package have been pub- lished, for instance, byOrabi (2016),Mosorov et al. 2016andGallardo et al. (2015).
3.1. Example #1
Thefirst simulated example is aflow rate measurement basing on gamma radiation measuring. The role of the moving source is played by the radioactive particle which is injected into an investigatedflow so that the pulse of intensity moves with theflow to be measured. Such a method is called pulse velocity method (Charlton, 1986). Typically in such measurements the mutually delayed stochastic signals are sensed by detectors located on the outer walls of the installation (Petryka et al., 2016). The measured time delay of signals is applied to determine the velocity of the marked phase. The method is based on the assumption that the trajectory of the particle is linear during all the measurements i.e., the investigatedflow has laminar character. In practice, this as- sumption can be fulfilled because two detectors are typically located closely enough.
The simulated equipment for pulse-velocity measurement of aflow is shown inFig. 3. It is assumed that the simulated column is a PVC column (density = 1.38 g/cm−3) with inner radius equal to 7.5 cm and wall thickness equal to 0.5 cm. The column is filled up with distilled water at room temperature and under atmospheric pressure. Two 2″× 2″ NaI(Ti) scintillation detectors (L/D = 1 and density crystal = 3.67 g/cm3) were positioned by the column wall. The detectors are placed on two planes and the distance between the planes is 9 cm.
Isotope Sc-46 was chosen as the radioactive particle. Sc-46 emits gamma-ray energies of 0.889 and 1.120 MeV. In simulations, these two gamma rays are modelled by a gamma radiation source with average energy equal to 1.004 MeV. The activity of the particle is set to 20 mCi which means the source will emit 1.48·109photons/s. Point description was also selected to model the source since the source diameter can equal to 3 mm. In the simulations counting threshold was determined by the photopeak and set to 0.95 MeV. The threshold value of countsNf
was set to 50.
Assuming that the distance between two detector planes is known and counts measured by thefirst detector are the same as the counts measured by second one, the delay time between two pulse series will allow to determine the time delay and next the particle velocity.
Forfixed time sampling, the time delay accuracy between two sig- nals generated by the detectors is determined by time samplingTand it is equal to 1 ms. Contrary to this classical approach, the accuracy of the delay time estimation based on the registration of intervalsTxwill be higher and it will be equal approximately to 0.2 ms (seeFig. 4b). This means that theflow velocity can be calculated more accurately because the time delay between two pulse series will be determined more ac- curately.
3.2. Example #2
Another example which illustrates the potential application of the proposed modified law is volumetric component fraction measurement.
According to classical approach, the component fraction can be derived from measurements of the linear attenuation coefficient (see Eq.(1)) of
a homogeneous mixture of two components. More details concerning the method can be found, for instance, inLali et al. (1987). To show the application of the modified law, the simulation of two component traction measurement has been done.
The simulated gauge was configured to measure the component fractions in a pipe. It is assumed that the densities of two components are known through calibration measurements and that they do not change. This principle can be applied to liquid/gas, solid/gas or solid/
liquid systems. In our simulation, we use the measurement of the liquid volume fraction of the liquid/gas mixture. The simulated facility in- cludes column fragment, collimator #1, wherein has been put a disk gamma radiation source, collimator #2 and NaI(Ti) detector. The cross section of a simulated facility is shown inFig. 5. The volume of interest is a PVC column (density = 1.38 g/cm−3) with inner radius equal to 1.5 cm and wall thickness equal to 0.5 cm. The column isfilled up with different levels of distilled water at room temperature and under at- mospheric pressure. The 1 in. NaI(Ti) scintillation detector with L/D = 1 and density crystal = 3.67 g/cm3was positioned by the column wall and the distance between them was 13 cm.
Isotope Am-241 emitting 59.5 keV photons was chosen as the radioactive source. The strength of the source was equal to 5 mCi.
Collimated gamma rays were absorbed while passing through the column due to different levels of water. Thus, detector will generate the time-varying counts depending on the volume fraction of liquid in the column. These registered counts will allow us to determine the liquid volume fraction in the mixture.
Several different simulations were conducted to verify the proposed approach.Fig. 6shows count rate vs. liquid volume fraction forfixed time samplingT= 300 ms. To compare the proposed approach against the classical one, changes of liquid volume fraction in time were si- mulated (seeFig. 7). These changes ranged from 1% to 37%. The counts measured by a classical detector for the fixed sampling time T = 300 ms are shown inFig. 8a. It is obvious, that the temporal resolution of all measurements is the same.
The series of proper pulses generated by a hypothetical detector is presented inFig. 8b where thefixed countNfis equal to 850. The value Nf= 850 reflected the minimum of the counts registered by the de- tector for the maximum of liquid level. As can be noted, the shorter intervalsTxallow the registration of the changing of the level of the liquid volume fraction more quickly. For instance, changing the level of the liquid volume fraction from 1% up to 5% corresponds to time in- tervalTx= 160 ms, and this means that time resolution is approxi- mately twice higher.
4. Conclusion
The article presents the modified Lambert-Beer law in time domain form. The count rate measured by the detector is changed to the re- gistration of time intervals. It is shown that it allows to increase tem- poral resolution of a gauge without the necessity to use a stronger radioactive source. Additionally, the accuracy of the measurements will still befixed and will not depend on measuring the count rates. The proposed approach can be successfully utilized in several typical ap- plications of radiation measurement systems. The proposed approach assumes the development of a new electronic part for a detector but these modifications will require the implementation of minor changes in current types of detectors.
References
J.S. Charlton (Ed.), Radioisotope techniques for problem solving in industrial process plants, Leonard Hill, 1986.
Croft J.R., 1991. ALARA: from theory towards practice, Office for Official Publications of the European Communities, p. 220.
Gallardo, S., Querol, A., Ortiz, J., Ródenas, J., Verdú, G., Villanueva, J.F., 2015.
Uncertainty analysis in environmental radioactivity measurements using the Monte Carlo code MCNP5. Radiat. Phys. Chem. 116, 214–218.
Johansen, G.A., Jackson, P., 2004. Radioisotope Gauges For Industrial Process Measurements. John Wiley and Sons, NY.
Lali, A.M., Khare, A.S., Joshi, J.B., Eapen, A.C., Rao, S.M., Yelgaonkar, V.N., Ajmera, R.L., 1987. Liquid-phase axial mixing in two-phase horizontal pipeflow. Int. J. Multiph.
Flow 13 (6), 815–821.
Monte Carlo Code Group, 2016. (accessed 29 December, 2016)〈https://mcnp.lanl.gov/〉.
Mosorov, V., Zych, M., Hanus, R., Petryka, L., 2016. Modelling of dynamic experiments in MCNP5 environment. Appl. Radiat. Isot. 112, 136–140.
Orabi, M., 2016. Studying factors affecting the indoor gamma radiation dose using the
MCNP5 simulation software. J. Environ. Radioact. 165, 54–59.
Petryka L., Mosorov V., Zych M., Hanus R., Sobota J., Jaszczur M.,Świsulski D., 2016.
Improvement of radiotracers experiments in mass transfer processes, In: Proceedings of the International Conference EFM16: Experimental Fluid Mechanics, Czech Republic, pp. 589–595.
Roy, S., Larachi, F., Al-Dahhan, M.H., Dudukovic, M.P., 2002. Optimal design of radio- active particle tracking experiments forflow mapping in opaque multiphase reactors.
Appl. Radiat. Isot. 56, 485–503.