Open access peer-reviewed chapter

# Mathematical Expressions of Radon Measurements

Submitted: February 2nd 2020Reviewed: April 24th 2020Published: December 23rd 2020

DOI: 10.5772/intechopen.92647

## Abstract

The measurement of radon, thoron and their progeny concentrations also leads to the knowledge of the presence of radioactive elements, which are the sources of these elements such as Uranium-238 and Thorium-232. Using of Solid State Nuclear Tracks Detectors (SSNTDs) it is probably the most widely applied for long term radon measurements. In this chapter, we derived the most important mathematical relationships that researchers need in radon measurements to calculate such as average radon concentration, exhalation rate, equilibrium factor, radon diffusion coefficient and transmission factor to get actual radon concentration in air atmosphere. The relationship between theoretical and experiment calibration drive and other mathematical relationships are given in this chapter.

### Keywords

• SSNTDs
• mathematical expressions

## 1. Introduction

In this chapter, we will drive and discuss the theoretical formalism which used in the research work and a complete methodology. This chapter will include the detail description of drive the most important mathematical relationships used in technique radon measurements. The present work will help in understanding the status of indoor and outdoor radon, thoron and their progeny concentrations and status of the exhalation of these gases from soil. Classification of measurements is also included in this chapter. The necessary procedures and formulae involved in measuring the concentrations of radon, thoron and their progeny, the radioactivity content of samples along with the calculations of, exhalation rate, equilibrium factor, radon diffusion coefficient and other mathematical relationships are given in this chapter.

## 2. Closed-can technique measurements

### 2.1 The buildup of radon concentration equation

Since radon is produced continuously from decay of radium in natural decay chains of uranium, the rate of change of the number of radon atoms is determined by radon decay and generation of radon in the decay of radium present in closed can Figure 1. Since radium present as solid and radon as gas, in order to find the rate of change of the number of radon atoms in the air-filled pore space, assuming that [1, 4].

• The radium is only present in soil and decay there.

• The soil column is homogeneous.

• The radium distributes uniformly in the surface soil and does not exist in the air.

• The radon production and decay in the air space.

• Radon transport in the soil is vertical and only due to diffusion and convection in the pore space.

• All radon produced in the solid material will escape (emanate) into the pore-air space.

• Radon-tight containers, no leakage of radon out of the can and no back diffusion effects.

Therefore, decay of radium and production of radon can be described by the rate equations for serial radioactive decay chain (Batman equations) [5]:

dNRndt=λRaNRaλRnNRnE2

From Eqs. (1) and (2), we can determine the number of undecided radon atoms at a time t. By solving two equations, we will obtain the remaining number of radon atoms without decay at any time is:

NRnt=NRa0λRaeλRatλRnλRa+eλRntλRaλRnE3

or

NRnt=λRaλRnλRaNRa0eλRateλRntE4

NRa0, is the original number of radium atoms. The activity A(t) at time t is defined mathematically:

At=λNtE5

Eq. (5) becomes:

ΑRnt=λRnλRnλRaΑRa0eλRateλRntE6

ΑRa0is original activity of radium which is constant value.

Since T12RaT12RnλRaλRaeλRat1&λRnλRaλRn. Therefore, Eq. (6) becomes:

ΑRnt=ΑRa01eλRntE7

From Eq. (7) the activity of radon grows and becomes exactly the same with original activity of radium when time t passed many of the radon half-life (i.e. 27days). In other word, radon atoms are decaying at the same rate at which they are formed. This is called secular equilibrium [5]. The secular equilibrium is important for the calculation of the activity concentration of radon in the can technique. This means that, the radon activity will reach maximum value or steady state value or equilibrium state value after 4 weeks time. This value is called sometimes the final activity or the saturated activity. In other words, we replaced ΑRa0by steady state (final) activity of radon Asin Eq. (7) to become:

ARnt=As1eλRntE8

Eq. (8) describes the buildup of radon activity through time t. If V is the volume of air-filled space within can, the activity concentration of radon CBq·m3in the air volume of the can given by the following relation [2, 3]:

C=ARnV=λRnNRnVE9

Eq. (9) becomes:

Ct=Cs1eλRntE10

Where Ctis the radon concentration at time t Bq·m3, Csis the steady state (final) concentration Bq·m3. Eq. (10) is the well-known equation which describes the buildup of the concentration of radon emanated from each sample inside the exhalation container with time [1, 6].

### 2.2 Track density-radon concentration relation

Since the alpha particles emitted by 222Rn and its progeny strike the detectors and leave latent tracks in it, Solid State Nuclear Detector measures the total number of alpha-disintegration in unit volume of the can during the exposure time. The tracks can be visible by chemical or electrochemical etching. The main measured quantity is the track density, which is the total tracks per unit area of detector, i.e. [1].

where ρis the track density expressed in Track·cm2, ADis the area of detector in (cm2). Since the etching track is observed and accounted by using optical microscope, and when looking into a microscope, we will see a lit circular area called field of view. The field of view (FOV) is the maximum area visible through the lenses of a microscope, and it is represented by a diameter. Therefore, we divide the area of detector to nof the field of view AFOV. Eq. (11) becomes:

ρ=Ntotal tracknAFOV=NavgAFOVE12

where nthe number of fields is, Navgis the average of total tracks and AFOVis the area of the field of view (cm2). The measured track density rate recorded on the SSNTD is proportional to the radon concentration during the time of exposure [7].

dtCtE13

or

dt=KCtE14

The proportionality constant, is called the calibration factor of the detector or conversional factor or response factor or turned over sensitivity factor [1]. It convert the track density (Track·cm2) to exposure concentration (Bq · m−3 · day). By integrate Eq. (14), we obtain with initial conditionρ0=0:

ρ=K0TCtdtE15

where Ctis radon concentration in air around the detector (Bq · m−3) at time t and T is the total exposure time (day). There are five cases, we can discuss than blow:

1. The radon concentration is proportion to along exposure time. In this state, the track density measured the integrated concentration and not concentration instantaneous or the final concentration. Sometimes, it is called accumulation concentration or exposure concentration. The integrated radon concentration CI(Bq·m3· day) after a period of time T is defined mathematically as [1]:

CI=0TCtdtE16

From Eqs. (15) and (16), we obtain:

CI=ρKE17

1. We define the average radon concentration (Bq · m−3) by the expression of:

Cavg=1T0TCtdtE18

From Eqs. (15) and (18), we get:

Cavg=ρKTE19

Cs=ρKT27dE20

This means that, the detector should be exposed for at least 27 day to record tracks for radon concentration.

The results lead to the following remarks [1]:

1. The radon concentration reaches the equilibrium state at the same time (t7T1/2) regardless of the volume of the container.

2. There are three different quantities measured by track density, integrated radon concentration, average radon concentration and saturate radon concentration as shown in Table 1.

3. The average radon concentration equal to the saturate radon concentration at large exposure time.

Integrated concentration CI(Bq · m−3 · day)Average concentration Cavg (Bq · m−3)Steady state concentration Cs (Bq · m−3)
ρKρKTρKT27d

### 2.3 Radon exhalation rate equation

dNtdt=EAλNtE21

where N(t) is the total number of radon atoms present in the can at time t, E is the exhalation rate (Bq · m−2 · h−1), A is cross-sectional area of the can (the surface area of sample from which the exhalation takes place), λ is the decay constant of radon (h−1) and τis the live time of radon (h). The quantity τdoes not exist in equation of reference [2, 3], that we added to the equation Eq. (21) to unify the units on both sides of the equation. The solution of Eq. (21) with initial condition N(0) = 0 is Eq. (22):

Nt=EAλ1eλtE22

If V is the volume of air (m3), the activity concentration of radon C(t) in the air volume of the can as a function of time t that can be given by the following relation Eq. (23):

Ct=AtV=λNtVE23

The equation becomes:

Ct=EAAτV1eλtE24
Butτh=1λh1E25
Ct=EAAλV1eλtE26

By integrating on sides of equation for time, we get:

CI=EAλVTeffE27

So, the exhalation radon rate as a function radon integrated concentration is given by:

EA=CIλVATeffE28

At steady state (secular equilibrium), dNtdt=0andTeff=T, we get:

EA=CsλVAE29

And the exhalation radon rate as a function average radon concentration is given by:

EA=CavgλVTATeffE30

From Eqs. (19) and (30), we get the exhalation rate as a function to the tracks density:

EA=ρλVKATeffE31

By the same method we get the mass exhalation rate, where M is the mass of sample in container:

E32

ΑRnt=ΑRa1eλRntE33

By dividing Eq. (33) on the volume of radon, we get on the activity concentration of radon:

CRnt=ΑRaV1eλRntE34

By multi Eq. (34) by dt and integrated for exposure time T, we get:

ΑRa=CIVTeffE35

By dividing Eq. (35) on mass of sample, we get:

CRa=CIVMTeffE36

where CRais effective radium content in unite (Bq/kg):

CRa=ΑRaME37

By average radon concentration, we get:

CRa=CavgTVMTeffE38

From Eqs. (19) and (38) we get the effective radium content as a function to the tracks density:

CRa=ρAhKMTeffE39

Since the unit of time in exhalation formula in hour, we should convert units of λ, Teffand K to hour. All the quantities in Table 2 are known except (CI, C, Cavg ). We can find ρexperimentally by using CR-39 or LR-115 type II based on radon dosimeter and by using continuous radon monitor to determine the value of (CI, C, Cavg ). From Table 2, we show that the quantities (λ,T,Teff,K,M,A,V)are constants, therefore the relationship between radon exhalation rate and effective radium content with the quantities (CI, C, Cavg , ρ) are linear. From the relations in Table 2, we get some relations which show the ship linear between these quantities [1]:

QuantityCICsCavgρ
EABqm2h1=CIλVATeffCsλVACavgTλVATeffρλVKATeff
EMBqkg1h1=CIλVMTeffCsλVMCavgTλVMTeffρλVKMTeff
CRaBqkg1=CIVMTeffCsVMCavgTVMTeffρVKMTeff

E40

### 2.5 Closed-can technique (two different detectors)

In this technique using two different SSNTD detectors (CR-39 & LR-II) were separately placed in close-can, to measurement and discriminate between radon and thoron concentrations which escape from sample, at same time Figure 2. During this exposure the α-particles emitted by the nuclei of radon and thoron and its progenies have bombarded the SSNTD films [1, 8].

The global track density rates, due to α-particles emitted by radon, registered on the CR-39 detector ρRnGcan be writing mathematical:

ρRnG=ρRn+ρPo218+ρPo214E41

where ρiare the track density for radon and its progenies on CR-39 detectors and from Eq. (19) can written by concentration average of radon and its progenies:

ρRnG=KRnCRnT+KPo218CPo218T+KPo214CPo214TE42

where Kiare the calibration factors for radon and its progenies. By malty and dividing right hand-Eq. (42) CRn, we obtain:

ρRnG=KRn+KPo218CPo218CRn+KPo214CPo218CRnCRnTE43

CPo218CRn=CPo214CRn=1E44

Eq. (43) becomes:

ρRnG=KRnCCRnTE45

where

KRnC=KRn+KPo218+KPo214E46

By same procedure, we find ρThGfor thoron on CR-39 detector. The total track density ρCRfor radon and thoron become:

ρCR=ρRnG+ρThG=KRCCRnT+KTCCThTE47

The final result, we obtain tow equation description relationship between the radon and thoron concentration and tack density on CR-39 and LR-II detectors:

E48

This are tow algebraic liner equations by tow variables CRnand CTh, the general solution for this are:

E49

After performing a series of mathematical simplifications, we obtain mathematical relationships to calculate concentrations for radon and thoron as function to tracks density ρr, radon-thoron concentrations Crand calibration factors (KL,KRn,KTh,KL) ratio.

E50
E51
E52

where:

ρCR: Tracks density registration on CR-39 detector.

ρLR: Tracks density registration on LR-115(II) detector.

KRC : Calibration factor of radon for CR-39 detector.

KRL : Calibration factor of radon for LR-115(II) detector.

KTC : Calibration factor of thoron for CR-39 detector.

KTL : Calibration factor of thoron for LR-115(II) detector.

Since, Cris positive quantity, the CR-LR track density ratio should be KRn<ρr<KThand KTh>KRn. Since, ρCR>ρLR, because CR-39 recorders all alpha particles energies, while LR-II detector recorders window energies (i.e. between Emin and Emax), this mean ρr>1[9].

## 3. Indoor radon technique measurements

Radon and thoron present in outdoor and indoor air as they exhaled from soil and building materials of the walls, floor and ceilings. It is critically important that inhalation of radon and their progeny concentrations has been shown experimentally to cause lung cancer in rats and observed to cause lung cancer in men exposed to large amounts in the air of mines. Several techniques have been developed to measure radon indoors. The use of a Solid State Nuclear Track Detector (SSNTD) closed in a cup mode (passive dosimeters) and bare mode has turned out to be the most appropriate for long term measurements. Radon measurement with, bare (open) detector are rough and with rather high uncertainties. Some device of cup mode have single chamber called invented cup. In twin cup mode, the detectors are kept inside a twin cup dosimeter, a cylindrical plastic chamber divided into two equal compartments. The two equal compartments on both sides are filter and pinhole compartments. There is one small compartment at the external middle attached to it which is used for bare mode exposure as shown in Figure 3 [1, 10, 11].

This design of the dosimeter was well suited to discriminate 222Rn and 220Rn in mixed field situations, where both gases are present. SSNTDs were used as detectors and affixed at the bottom of each cup as well as on the outer surface of the dosimeter. The exposure of the detector inside the cup is termed as cup mode and the one exposed open is termed as the bare mode. One of the cups had its entry covered with a glass fiber filter paper that permits both 222Rn and 220Rn gases into the cup and is called the filter cup. The other cup was covered with a soft sponge and is called the sponge cup. SSNTDs film inside the sponge cup registers tracks contributed by 222Rn only, while that in the filter cup records tracks due to 222Rn and 220Rn. The third SSNTDs film exposed in the bare mode registers alpha tracks contributed by the concentrations of both the gases and their alpha emitting progenies. The dosimeters were kept at a height of 1.5 m from the ceiling of the room and care was taken to keep the bare card at least 10 cm minimum away from any surface. This ensured that errors due to tracks from deposited activity from nearby surfaces were avoided, since the ranges of alpha particles from 222Rn/220Rn are about 10 cm. The global track density rates, due to α-particles emitted by radon, registered on the SSNTDs can write mathematical [11]:

ρS=KRSTCRnE53
ρF=KRFTCRn+KTFTCThE54

The general solution for this are:

CRn=ρSKRSTE55
CTh=ρSKTFTρFρSKRFKRSE56

where:

ρS= Tracks density for sponge cup.

ρF= Tracks density for filter piper cup.

KRS =Calibration factor of radon for sponge cup.

KRF  = Calibration factor of radon for filter piper cup.

KTF  = Calibration factor of thoron for filter piper cup.

Since, CThis positive quantity, ρFρStrack density ratio should be ρFρS>KRFKRS.

## 4. Equilibrium factor

The equilibrium factor (F) is defined as the ratio of potential alpha energy concentration (PAEC) of actual air-radon mixture (also radon progeny) to the PAEC in secular equilibrium with radon. The equilibrium factor (F) is an important parameter in the determination of radon equivalent dose. A common practice for radon hazard assessment nowadays is to, first, determine the radon gas concentration and then to apply an assumed F with a typical value of about 0.4 [12]. However, in reality, F varies significantly with time and place, and an assumed F cannot reflect the actual conditions. Actually, the radon concentration was very dependent on the ventilation rate, therefore F was determined by SSNTD based on using can and bare method. In this method, two similar detectors were exposed to radon, one in can-mode configuration (in can detector) and the other in bare-mode configuration (bare detector). F can be found as a function of the track density ratio ρBS between bare (ρB ) and in can (ρS ) detector, respectively. There are three steps for calculating the equilibrium factor [1, 11, 13]:

1. The potential alpha energy concentration of any mixture of (short-live) radon or thoron daughters in air.

2. The working level (WL).

3. The ventilation rate as a function of the track density ratio between bare mode and can mode.

4. The equilibrium factor as a function of the ventilation rate.

The potential alpha energy concentration (PAEC) of any mixture of (short-live) radon or thoron daughters in air is the sum of the potential alpha energy of all daughters atoms present per unit volume of air. The usual unit for this quantity is MeV · l−1. This unit is related to the SI units J and m3 according to 1 J · m−3 = 6.24 × 109 Mev · l−1 [12]. To express mathematically, we let:

PAECTotal=PAEC1N1+PAEC2N2+PAEC3N3+PAEC4N4E57

where PAECTotal, is the total potential alpha energy concentration of any mixture of (short -live) radon daughters in air. PAECi(i = 1, 4) are the potential alpha energy for 218Po, 214Pb, 214Bi and 214Po, respectively. Ni is the number atoms of daughters of radon. The concentration potential alpha energy is defined as:

CPAET=PAETVE58
Ci=AiV=λiNiVE59

From Eqs. (57), (58) and (59), we get:

CPAET=PAE1λ1f1+PAE2λ2f2+PAE3λ3f3+PAE4λ4f4C0E60
E61

where fi, is the activity concentration fraction. A special unit for this quantity used for radiation protection purposes is the working level (WL). A WL is defined as corresponding approximately to the potential alpha energy concentration of short-live radon daughters in air which are in radioactive equilibrium with a radon concentration of 3.7 kBq · m−3 [12].

C1=C2=C3=C4=C0=CRn=3.7kBq.m3=3.7Bq·l1E62

Eq. (60) become as:

CPAECT=WL=3.7PE1λ1+3.7PE2λ2+3.7PE3λ3+3.7PE4λ4=1.3105Mevl1E63

The numerical calculations of Eq. (63) are shown in Table 3.

Daughters of radonα-energy (Mev)Half-life (s)Ultimate alpha energy (Mev)Total energy (Mev)Fraction energy
Po-21861836 + 7.691.34E+040.11
Pb-214zero1563zero+7.686.41E+040.51
Bi-214zero1182zero +7.684.85E+040.38
Po-2147.691.64E-047.686.72E-03zero
CPAECT=1.3E+05

### Table 3.

Numerical calculations of Eq. (63).

One (WL) equal to 1.3 × 105 Mev · l−1 of air. To obtain the concentration potential alpha energy by fraction energy, Eq. (63) multi and divide by 3.7 Bq · l−1 and multi and divide by 1.3 × 105 Mev · l−1:

CPET=3.7PE1λ11.3105f1+3.7PE2λ21.3105f2+3.7PE3λ31.3105f3+3.7PE4λ41.3105f41.31053.7CRnE64
CPET=0.11f1+0.51f2+0.38f31.31053.7CRnE65

To obtain the concentration potential alpha energy by unit WL is:

CPETWL=FRnCRnBql13.7E66

or

CPETmWL=FRnCRnBqm33.7E67

where

FRn=0.11f1+0.51f2+0.38f3E68

For Thoron daughters we find the concentration of thoron which give potential energy of alpha its daughters equal to 1.3 × 105 Mev · l−1 is 0.275 Bq · l−1, as shown in Table 4.

Daughters of thoronα-energy (Mev)Half-life (s)Ultimate alpha energy (Mev)Total energy (Mev)CTn (Bq · l−1)
Po-2166.780.1514.59523.16E+000.275
Pb-212zero38,3047.81524.32E+05
Bi-2126.136367.81524.10E+04
Po-2128.783.04E-078.783.85E-06
4.73E+05

### Table 4.

Numerical calculations of thoron concentration as corresponding to the total energy 1.3 × 105 Mev · l−1.

By the same setup, we drive Eq. (60) for thoron and numerical calculations of it, as shown in Table 5.

Daughters of thoronα-energy (Mev)Half-life (s)Ultimate alpha energy (Mev)Total energy (Mev)Fraction energy
Po-2166.780.1514.59528.69E-01zero
Pb-212zero38,3047.81521.19E+050.91
Bi-2126.136367.81521.13E+040.09
Po-2128.783.04E-078.781.06E-06zero
1.3E+05

### Table 5.

Numerical calculations of Eq. (63) for thoron.

The concentration potential alpha energy for thoron by unit WL is:

CPETmWL=FTnCTnBq.m30.275E69

where

FTn=0.91f2+0.09f3E70

FRnand FTnare the equilibrium factor for radon and thoron, respectively. The maximum value of equilibrium factor is F = 1, when the radon or thoron progeny are present in complete equilibrium with radon/thoron that is present. The minimum value of equilibrium factor is F = 0, that is mean no-equilibrium between the radon or thoron and its progeny. In our work, we measured the equilibrium factor depending on the ventilation rate. Ventilation rate is one of the parameters used to describe the perturbation caused in radioactive equilibrium of radon/thoron and its descendants in air. Decay of radon and production of radon can be described by the rate equations for serial radioactive decay chain (Batman equations):

dNidt=λi1Ni1ΛiNii=14E71

where

Λi=V+Ai+Wi+λiE72

The first term on the right is the rate of formation of the ith-member of the progeny by radioactive decay of the (i-1)th-member, with constant λi1; the second term describes the radioactive leakage rate, owing to ventilation V, to aerosol grains Ai and deposition on the walls Wi to which it added the rate of radioactive decay of the ith-member of the progeny, λi. The index i, running from 1 to 4, labels the relevant daughter in the radon family: 218Po, 214Pb, 214Bi, 214Po. 222Rn itself will be label with i = 0. For thoron family: 216Po, 212Pb, 212Bi, 212Po. 220Rn (Tn) itself will be labeled with i = 0. Ventilation rate affects equally all members of the family. When steady state is reached, radon daughter’s activities Eq. (71) become as [13]:

λi1Ni1=ΛiNiE73

or

CiCi1=λiΛi=diE74

From Eq. (74), we obtain:

E75

where C4C3=d4=1since C3=C4Forλ3λ4

When the ventilation rate is the only environmental affecting disequilibrium or when it is the dominant on, ventilation rates and the equilibrium factor are obtained by track density measurements, so:

Λi=V+λiE76

The track density of both bare mode and can mode (with sponge filter) detector relates the concentration of radon and its daughters as:

In can mode:

ρS=ρ0+ρ1+ρ4E77
ρS=K0C0T+K1C1T+K4C4TE78
ρS=K0+K1f1+K4f4C0TE79

In can mode f1=f2=f3=f4=1, Eq. (79) become as:

ρS=KSC0TE80

where

Ks=K0+K1+K4E81

In bare mode (in absence thoron)

ρB=ρ0+ρ1+ρ4E82
ρB=K¯0+K¯1f1+K¯4f4C¯0TE83

In bare mode f1f2f3f4because no equilibrium between the radon and its progenies, but K¯0=K¯1=K¯4=KB, therefore Eq. (83) become as:

ρB=KB1+f1+f4C¯0TE84

By dividing Eq. (84) on Eq. (80) when C¯0=C0, we obtain:

1+f1+f4=KSBρBSE85
E86

From Eqs. (74), (75), (76) and (84), we obtain:

λ1λ1+V+λ1λ1+Vλ2λ2+Vλ3λ3+V=KSBρBS1E87

After some setup, we obtain:

V3+a2V2+a1V+a0=0E88

where

E89

Ventilation rate is the solution of Eq. (89), obtainable by means of standard algebraic procedures. The equilibrium factor is strongly dependent on the ventilation rate. This dependence was expressed by using Eqs. (68), (70), (75) and (76).

FRn=λ1λ1+V0.11+0.51λ2λ2+V+0.38λ2λ3λ2+Vλ3+VE90
FTn=λ1λ2λ1+Vλ2+V0.91+0.09λ3λ3+VE91

Values of the equilibrium factor F, follow from solution of Eq. (88) and combined Eq. (90) and Eq. (91).

## 5. Determination of the radon diffusion coefficient in porous medium

The container 1 has volume V1 and radon concentration C1 and container 2 has volume V2 and radon concentration C2 . It is assumed that the material represents thickness d and area A produced P is the radon production rate (Bq · m−3 · s−1) and has diffusion coefficient D. We neglect radon production within the porous material itself and no leakage in containers. Radon transport in porous material is described by a general equation of continuity, which includes four basic processes: generation, decay, diffusion and convection. Under the supposition of the time-dependent one-dimensional differential equation (no convection) describing the radon activity concentration C0 is given by Ficks second law:

C0ztt=De2C0ztz2λC0zt+PE92

where C0ztis the radon concentration within the pores (Bq/m3), D is the bulk diffusion coefficient (m2/h, gas flux expressed per unit area of material as a whole), De is the effective diffusion coefficient λis the radon decay constant, εis the prostiy and P is the production rate of radon in the pore air (Bq/m3 · h):

De=DεE93
P=fλCRaρεE94

where, fis the emanation fraction, ρis the bulk density (kg/m3) and CRais the radium concentration (Bq/kg).

In steady state C0ztt=0and Eq. (92) becomes:

d2C0zdz2λDeC0z+PDe=0E95

The general solution is:

C0z=Aez/l+Bez/l+PλE96
l=DeλE97

where A, B are integral constant obtained by boundary conditions, lis the diffusion length of radon (m). The boundary conditions are formed by, it is assumed that the radon concentration at the boundary conditions is specified by the radon concentration of incoming air and the radon concentration is set to a constant value at these boundary conditions. Two boundary conditions at equilibrium rate, we needed:

E98

We get two equations:

C1=A+B+PλE99
C2=Aeβ+Beβ+PλE100

where

β=dlE101

By solving Eqs. (99) and (100) and some mathematical step, we get:

A=eβeβeβC1+1eβeβC21eβeβeβPλE102
B=eβeβeβC11eβeβC2+1eβeβeβPλE103

By using the relations:

E104

And mathematical steps, we get the general solution:

C0z=sinhdzlsinhdlC1+sinhzlsinhdlC2+sinhdzl+sinhzlsinhdl1PλE105

E=λVC2AE106

From Ficks first law:

E=DedC0zdzz=dE107

We get:

dC0zdz=coshdzllsinhdlC1+coshzllsinhdlC2+coshzαcoshdzαlsinhdlPλE108
dC0zdzz=d=1lsinhdlC1+coshdllsinhdlC2+coshdα1lsinhdlPλE109

Substitute Eq. (109) in Eq. (107), we obtain:

E=DelsinhdlC1+coshdlC2coshdl1PλE110

From Eq. (106) and Eq. (110), we get:

C2=C1coshdl+λlV2DeAsinhdl+1coshdlcoshdl+λlV2DeAsinhdlPλE111

Since,

E112

Substitute set Eq. (112) in Eq. (111), we get:

C2=C1coshdl+hlsinhdl+1coshdlcoshdl+hlsinhdlPλE113

Rewrite Eq. (113):

1PλC2coshdl+hlsinhdlC1C2PλC2=0E114

From Eqs. (101) and (114), we get final relation:

1PλC2coshβ+hdβsinhβC1C2PλC2=0E115

Eq. (115) is nonlinear equation; therefore, the radon diffusion coefficient D can be numerically calculated by using the Newton-Raphson method.

IF βis small, and by using set relation in Eq. (116), we can make approximation to evolution radon diffusion coefficient by simple relation as:

E116

We find simple relation to calculate the effective diffusion coefficient Eq. (117).

De=λdhC1C21E117

## 6. Transmission factor

The purpose of using covers of cup (sponge or paper filter) to allow entry radon and thoron gases only without its daughters for filter paper and radon only for sponge. In actual terms it means that the covers are reduced from concentration inter cup to detection [16]. Therefore, we define transmission factor as the ratio between the concentrations inter cup and outside cup. The purpose from calculate diffusion coefficient and transmission factor to get actual radon concentration in air atmosphere. It is assumed that the gas enters the chamber through porous filter by the process of diffusion with diffusion coefficient D1and D2in air. If C1 is the average radon/thoron gas concentration in the cup and Co is out the cup. Let the radon diffusion in x-direction Figure 5 [1, 14, 16].

Then, the steady state equations for described the radon gas diffusion internal porous filter and in air space in cup may be written as Eq. (95), (P = 0, because of the lack of a source of radium) respectively:

d2C1xdx2λD1C1x=0E118
d2C2xdx2λD2C2x=0E119

The general solutions are

C1x=A1ex/l1+B1ex/l1E120
C2x=A2ex/l2+B2ex/l2E121
E122

where A1, B1, A2, B2, are integral constants obtained by boundary conditions, l1,l2are the diffusion length of radon (m), D1, D2 are diffusion coefficients for porous filter and air respectively.

1. The boundary conditions at equilibrium at sides of porous filter are:

E123

The solutions of these equations are:

E124

From Eqs. (120) and (124), we get the general solution is:

C1x=sinhxδl1sinhδl1Ci+sinhxl1sinhδl1CoE125

1. The boundary conditions at equilibrium in air space enter cup are:

E126

We get two equations:

E127

The solutions of these equations are:

E128

From Eqs. (121) and (128), we get the general solution as:

C2x=sinhxLl2sinhLl2CiE129

The average concentration radon inter cup, we get from relation:

C¯=1L0LC2xdxE130

The average concentration radon inter cup:

C¯=l2LtanhLl2CiE131

1. The boundary condition number 3 is:

E1x=0=E2x=0E132

or

E133

and

dC2xdxx=0=1l2tanhLl2CiE134

Substitute Eq. (134) in Eq. (133), we get:

Ci=Cocoshδl1+D2D1l1l2sinhδl1tanhLl2E135

IF δl1is small sinhδl1=δl1and coshδl1=1, we get:

C¯=l2Lλl2δD1+cothLl2CoE136

From define transmission factor we get as:

Transmission%=C¯Co=l2Lλl2δD1+cothLl2E137

## 7. The calibration factor

Quantitative measurements with both single and multi-SSNT detectors device can be performed only if the calibration factor is known. It can be measured experimentally or estimated theoretically. Moreover, the theoretically derived calibration factor relations may supply reasonable basic design criteria. There are many method to deriving formulas of calibration factor such as Mont- Carlo, mean critical angle by unit cm. At the same time, the radon chamber is used to determine the calibration factors for different dosimeter geometry configuration [1, 17, 18, 19]. The theoretical calibration factor is calculated by unit (cm), while the experiment calibration factor is measured by Trackcm2Bq·m3·day. Since 1 Bq = Disintegration/second = track/second, 1 day = 86,400 sec and 1 m3 = 106 cm3, so, 1 cm = 0.0864 Trackcm2Bq·m3·day[20]. The relation between Ktheoreticaland Kexperementin the can technique is:

KexperementTrackcm2Bq·m3·day=0.0864iKitheoreticalcmE138

## 8. Conclusion remarks

1. The concentration of radon emanated from each sample inside the closed container reach to the steady state after a fixed period of time without leakage out of the container Eq. (10).

2. The detector should be exposed for at least 27 day to record tracks for radon concentration.

3. The radon concentration reaches the equilibrium state at the same time (t ≅ 7T_(1/2)) regardless of the volume of the container.

4. There are three different quantities measured by track density, integrated radon concentration, average radon concentration and saturate radon concentration as shown in Table 1.

5. From the relations in Table 2, we get some relationship between radon concentration, area and mass radon exhalation and effective radium content are liner equations Table 2.

6. The radon/thoron concentrations equations which used in technique two different SSNTDs are very critical because it’s a mathematical relationship as a fraction.

7. For more accurate estimation of the effective dose from radon/thoron exposure, one should measure the equilibrium factor at each site.

8. High dose does not necessarily mean there is a high concentration of radon, there may be high equilibrium factor (bad ventilation), and so we recommend interest in ventilation factor when houses and buildings design.

9. It is necessary to correct the radon concentration when using covers (sponge or filter paper) in measurements of radon concentration by transmission factor.

## Acknowledgments

I would like to express my deep appreciation to:

• My supervisor Prof. Dr. Abdul Ridha Hussein Subber

• Department of physics

• Education College for Pure Science, Basra University

• Oil Ministry, Gas Filling Company and My Company (Basra Oil)

• Dr. Basim Almayahi and IntechOpen publishing

My great thanks to all which they could assistance me in any way.

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Ali Farhan Nader Alrekabi (December 23rd 2020). Mathematical Expressions of Radon Measurements, Recent Techniques and Applications in Ionizing Radiation Research, Ahmed M. Maghraby and Basim Almayyahi, IntechOpen, DOI: 10.5772/intechopen.92647. Available from:

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