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# Mathematical Expressions of Radon Measurements

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Submitted: 02 February 2020 Reviewed: 24 April 2020 Published: 23 December 2020

DOI: 10.5772/intechopen.92647

From the Edited Volume

## Recent Techniques and Applications in Ionizing Radiation Research

Edited by Ahmed M. Maghraby and Basim Almayyahi

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## 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]:

d N Ra dt = λ Ra N Ra E1
d N Rn dt = λ Ra N Ra λ Rn N Rn E2

where λ Ra , λ Rn is decay constant for radium and radon respectively. Τ 1 / 2 Ra = 1600 years , Τ 1 / 2 Rn = 3.824 d ays are the half-life for radium and radon respectively.

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:

N Rn t = N Ra 0 λ Ra e λ Ra t λ Rn λ Ra + e λ Rn t λ Ra λ Rn E3

or

N Rn t = λ Ra λ Rn λ Ra N Ra 0 e λ Ra t e λ Rn t E4

N Ra 0 , is the original number of radium atoms. The activity A(t) at time t is defined mathematically:

A t = λ N t E5

Eq. (5) becomes:

Α Rn t = λ Rn λ Rn λ Ra Α Ra 0 e λ Ra t e λ Rn t E6

Α Ra 0 is original activity of radium which is constant value.

Since T 1 2 Ra T 1 2 Rn λ Ra λ Ra e λ Ra t 1 & λ Rn λ Ra λ Rn . Therefore, Eq. (6) becomes:

Α Rn t = Α Ra 0 1 e λ Rn t E7

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. 27 days ). 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 Α Ra 0 by steady state (final) activity of radon A s in Eq. (7) to become:

A Rn t = A s 1 e λ Rn t E8

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 C Bq · m 3 in the air volume of the can given by the following relation [2, 3]:

C = A Rn V = λ Rn N Rn V E9

Eq. (9) becomes:

C t = C s 1 e λ Rn t E10

Where C t is the radon concentration at time t Bq · m 3 , C s is the steady state (final) concentration Bq · m 3 . 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].

ρ = N total track A D E11

where ρ is the track density expressed in Track · cm 2 , A D is 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 n of the field of view A FOV . Eq. (11) becomes:

ρ = N total track n A FOV = N avg A FOV E12

where n the number of fields is, N avg is the average of total tracks and A FOV is 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].

dt C t E13

or

dt = K C t E14

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 · cm 2 ) to exposure concentration (Bq · m−3 · day). By integrate Eq. (14), we obtain with initial condition ρ 0 = 0 :

ρ = K 0 T C t dt E15

where C t is 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 C I ( Bq · m 3 · day) after a period of time T is defined mathematically as [1]:

C I = 0 T C t dt E16

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

C I = ρ K E17

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

C avg = 1 T 0 T C t dt E18

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

C avg = ρ K T E19

C s = ρ K T 27 d E20

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 ( t 7 T 1 / 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 C I (Bq · m−3 · day) Average concentration Cavg (Bq · m−3) Steady state concentration Cs (Bq · m−3)
ρ K ρ K T ρ K T 27 d

### 2.3 Radon exhalation rate equation

dN t dt = E A λ N t E21

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 τ is the live time of radon (h). The quantity τ does 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):

N t = E A λ 1 e λt E22

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):

C t = A t V = λ N t V E23

The equation becomes:

C t = E A A τ V 1 e λt E24
But τ h = 1 λ h 1 E25
C t = E A A λV 1 e λt E26

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

C I = E A λV T eff E27

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

E A = C I λ V A T eff E28

At steady state (secular equilibrium), dN t dt = 0 and T eff = T , we get:

E A = C s λ V A E29

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

E A = C avg λ V T A T eff E30

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

E A = ρ λ V KA T eff E31

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

E32

Α Rn t = Α Ra 1 e λ Rn t E33

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

C Rn t = Α Ra V 1 e λ Rn t E34

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

Α Ra = C I V T eff E35

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

C Ra = C I V M T eff E36

where C Ra is effective radium content in unite (Bq/kg):

C Ra = Α Ra M E37

By average radon concentration, we get:

C Ra = C avg T V M T eff E38

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

C Ra = ρ A h K M T eff E39

Since the unit of time in exhalation formula in hour, we should convert units of λ , T eff and K to hour. All the quantities in Table 2 are known except (CI, C, Cavg ). We can find ρ experimentally 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 , T eff , 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]:

Quantity C I C s C avg ρ
E A Bq m 2 h 1 = C I λ V A T eff C s λ V A C avg T λ V A T eff ρ λ V KA T eff
E M Bq kg 1 h 1 = C I λ V M T eff C s λ V M C avg T λ V M T eff ρ λ V KM T eff
C Ra Bq kg 1 = C I V M T eff C s V M C avg T V M T eff ρ V K M T eff

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 ρ Rn G can be writing mathematical:

ρ Rn G = ρ Rn + ρ Po 218 + ρ Po 214 E41

where ρ i are 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:

ρ Rn G = K Rn C Rn T + K Po 218 C Po 218 T + K Po 214 C Po 214 T E42

where K i are the calibration factors for radon and its progenies. By malty and dividing right hand-Eq. (42) C Rn , we obtain:

ρ Rn G = K Rn + K Po 218 C Po 218 C Rn + K Po 214 C Po 218 C Rn C Rn T E43

C Po 218 C Rn = C Po 214 C Rn = 1 E44

Eq. (43) becomes:

ρ Rn G = K RnC C Rn T E45

where

K RnC = K Rn + K Po 218 + K Po 214 E46

By same procedure, we find ρ Th G for thoron on CR-39 detector. The total track density ρ CR for radon and thoron become:

ρ CR = ρ Rn G + ρ Th G = K RC C Rn T + K TC C Th T E47

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 C Rn and C Th , 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 ρ r , radon-thoron concentrations C r and calibration factors ( K L , K Rn , K Th , K L ) ratio.

E50
E51
E52

where:

ρ CR : Tracks density registration on CR-39 detector.

ρ LR : 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, C r is positive quantity, the CR-LR track density ratio should be K Rn < ρ r < K Th and K Th > K Rn . Since, ρ CR > ρ LR , because CR-39 recorders all alpha particles energies, while LR-II detector recorders window energies (i.e. between Emin and Emax), this mean ρ r > 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]:

ρ S = K RS T C Rn E53
ρ F = K RF T C Rn + K TF T C Th E54

The general solution for this are:

C Rn = ρ S K RS T E55
C Th = ρ S K TF T ρ F ρ S K RF K RS E56

where:

ρ S = Tracks density for sponge cup.

ρ F = 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, C Th is positive quantity, ρ F ρ S track density ratio should be ρ F ρ S > K RF K RS .

## 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:

PAEC Total = PAEC 1 N 1 + PAEC 2 N 2 + PAEC 3 N 3 + PAEC 4 N 4 E57

where PAEC Total , is the total potential alpha energy concentration of any mixture of (short -live) radon daughters in air. PAEC i (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:

C PAET = PAET V E58
C i = A i V = λ i N i V E59

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

C PAET = PAE 1 λ 1 f 1 + PAE 2 λ 2 f 2 + PAE 3 λ 3 f 3 + PAE 4 λ 4 f 4 C 0 E60
E61

where f i , 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].

C 1 = C 2 = C 3 = C 4 = C 0 = C Rn = 3.7 kBq . m 3 = 3.7 Bq · l 1 E62

Eq. (60) become as:

C PAECT = WL = 3.7 PE 1 λ 1 + 3.7 PE 2 λ 2 + 3.7 PE 3 λ 3 + 3.7 PE 4 λ 4 = 1.3 10 5 Mev l 1 E63

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

Daughters of radon α-energy (Mev) Half-life (s) Ultimate alpha energy (Mev) Total energy (Mev) Fraction energy
Po-218 6 183 6 + 7.69 1.34E+04 0.11
Pb-214 zero 1563 zero+7.68 6.41E+04 0.51
Bi-214 zero 1182 zero +7.68 4.85E+04 0.38
Po-214 7.69 1.64E-04 7.68 6.72E-03 zero
C PAECT = 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:

C PET = 3.7 PE 1 λ 1 1.3 10 5 f 1 + 3.7 PE 2 λ 2 1.3 10 5 f 2 + 3.7 PE 3 λ 3 1.3 10 5 f 3 + 3.7 PE 4 λ 4 1.3 10 5 f 4 1.3 10 5 3.7 C Rn E64
C PET = 0.11 f 1 + 0.51 f 2 + 0.38 f 3 1.3 10 5 3.7 C Rn E65

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

C PET WL = F Rn C Rn Bq l 1 3.7 E66

or

C PET mWL = F Rn C Rn Bq m 3 3.7 E67

where

F Rn = 0.11 f 1 + 0.51 f 2 + 0.38 f 3 E68

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 thoron α-energy (Mev) Half-life (s) Ultimate alpha energy (Mev) Total energy (Mev) CTn (Bq · l−1)
Po-216 6.78 0.15 14.5952 3.16E+00 0.275
Pb-212 zero 38,304 7.8152 4.32E+05
Bi-212 6.1 3636 7.8152 4.10E+04
Po-212 8.78 3.04E-07 8.78 3.85E-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 thoron α-energy (Mev) Half-life (s) Ultimate alpha energy (Mev) Total energy (Mev) Fraction energy
Po-216 6.78 0.15 14.5952 8.69E-01 zero
Pb-212 zero 38,304 7.8152 1.19E+05 0.91
Bi-212 6.1 3636 7.8152 1.13E+04 0.09
Po-212 8.78 3.04E-07 8.78 1.06E-06 zero
1.3E+05

### Table 5.

Numerical calculations of Eq. (63) for thoron.

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

C PET mWL = F Tn C Tn Bq . m 3 0.275 E69

where

F Tn = 0.91 f 2 + 0.09 f 3 E70

F Rn and F Tn are 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):

d N i dt = λ i 1 N i 1 Λ i N i i = 1 4 E71

where

Λ i = V + A i + W i + λ i E72

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 λ i 1 ; 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, λ i . 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]:

λ i 1 N i 1 = Λ i N i E73

or

C i C i 1 = λ i Λ i = d i E74

From Eq. (74), we obtain:

E75

where C 4 C 3 = d 4 = 1 since C 3 = C 4 For λ 3 λ 4

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:

Λ i = V + λ i E76

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:

ρ S = ρ 0 + ρ 1 + ρ 4 E77
ρ S = K 0 C 0 T + K 1 C 1 T + K 4 C 4 T E78
ρ S = K 0 + K 1 f 1 + K 4 f 4 C 0 T E79

In can mode f 1 = f 2 = f 3 = f 4 = 1 , Eq. (79) become as:

ρ S = K S C 0 T E80

where

K s = K 0 + K 1 + K 4 E81

In bare mode (in absence thoron)

ρ B = ρ 0 + ρ 1 + ρ 4 E82
ρ B = K ¯ 0 + K ¯ 1 f 1 + K ¯ 4 f 4 C ¯ 0 T E83

In bare mode f 1 f 2 f 3 f 4 because no equilibrium between the radon and its progenies, but K ¯ 0 = K ¯ 1 = K ¯ 4 = K B , therefore Eq. (83) become as:

ρ B = K B 1 + f 1 + f 4 C ¯ 0 T E84

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

1 + f 1 + f 4 = K SB ρ BS E85
E86

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

λ 1 λ 1 + V + λ 1 λ 1 + V λ 2 λ 2 + V λ 3 λ 3 + V = K SB ρ BS 1 E87

After some setup, we obtain:

V 3 + a 2 V 2 + a 1 V + a 0 = 0 E88

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).

F Rn = λ 1 λ 1 + V 0.11 + 0.51 λ 2 λ 2 + V + 0.38 λ 2 λ 3 λ 2 + V λ 3 + V E90
F Tn = λ 1 λ 2 λ 1 + V λ 2 + V 0.91 + 0.09 λ 3 λ 3 + V E91

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:

C 0 z t t = D e 2 C 0 z t z 2 λ C 0 z t + P E92

where C 0 z t is 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 λ is the radon decay constant, ε is the prostiy and P is the production rate of radon in the pore air (Bq/m3 · h):

D e = D ε E93
P = f λ C Ra ρ ε E94

where, f is the emanation fraction, ρ is the bulk density (kg/m3) and C Ra is the radium concentration (Bq/kg).

In steady state C 0 z t t = 0 and Eq. (92) becomes:

d 2 C 0 z d z 2 λ D e C 0 z + P D e = 0 E95

The general solution is:

C 0 z = A e z / l + B e z / l + P λ E96
l = D e λ E97

where A, B are integral constant obtained by boundary conditions, l is 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:

C 1 = A + B + P λ E99
C 2 = A e β + B e β + P λ E100

where

β = d l E101

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

A = e β e β e β C 1 + 1 e β e β C 2 1 e β e β e β P λ E102
B = e β e β e β C 1 1 e β e β C 2 + 1 e β e β e β P λ E103

By using the relations:

E104

And mathematical steps, we get the general solution:

C 0 z = sinh d z l sinh d l C 1 + sinh z l sinh d l C 2 + sinh d z l + sinh z l sinh d l 1 P λ E105

E = λ V C 2 A E106

From Ficks first law:

E = D e d C 0 z dz z = d E107

We get:

d C 0 z dz = cosh d z l l sinh d l C 1 + cosh z l l sinh d l C 2 + cosh z α cosh d z α l sinh d l P λ E108
d C 0 z dz z = d = 1 l sinh d l C 1 + cosh d l l sinh d l C 2 + cosh d α 1 l sinh d l P λ E109

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

E = D e l sinh d l C 1 + cosh d l C 2 cosh d l 1 P λ E110

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

C 2 = C 1 cosh d l + λl V 2 D e A sinh d l + 1 cosh d l cosh d l + λl V 2 D e A sinh d l P λ E111

Since,

E112

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

C 2 = C 1 cosh d l + h l sinh d l + 1 cosh d l cosh d l + h l sinh d l P λ E113

Rewrite Eq. (113):

1 P λ C 2 cosh d l + h l sinh d l C 1 C 2 P λ C 2 = 0 E114

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

1 P λ C 2 cosh β + h d β sinh β C 1 C 2 P λ C 2 = 0 E115

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

IF β is 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).

D e = λ d h C 1 C 2 1 E117

## 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 D 1 and D 2 in 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:

d 2 C 1 x d x 2 λ D 1 C 1 x = 0 E118
d 2 C 2 x d x 2 λ D 2 C 2 x = 0 E119

The general solutions are

C 1 x = A 1 e x / l 1 + B 1 e x / l 1 E120
C 2 x = A 2 e x / l 2 + B 2 e x / l 2 E121
E122

where A1, B1, A2, B2, are integral constants obtained by boundary conditions, l 1 , l 2 are 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:

C 1 x = sinh x δ l 1 sinh δ l 1 C i + sinh x l 1 sinh δ l 1 C o E125

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:

C 2 x = sinh x L l 2 sinh L l 2 C i E129

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

C ¯ = 1 L 0 L C 2 x dx E130

The average concentration radon inter cup:

C ¯ = l 2 L tanh L l 2 C i E131

1. The boundary condition number 3 is:

E 1 x = 0 = E 2 x = 0 E132

or

E133

and

d C 2 x dx x = 0 = 1 l 2 tanh L l 2 C i E134

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

C i = C o cosh δ l 1 + D 2 D 1 l 1 l 2 sinh δ l 1 tanh L l 2 E135

IF δ l 1 is small sinh δ l 1 = δ l 1 and cosh δ l 1 = 1 , we get:

C ¯ = l 2 L λ l 2 δ D 1 + coth L l 2 C o E136

From define transmission factor we get as:

Transmission % = C ¯ C o = l 2 L λ l 2 δ D 1 + coth L l 2 E137

## 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 Track cm 2 Bq · m 3 · day . Since 1 Bq = Disintegration/second = track/second, 1 day = 86,400 sec and 1 m3 = 106 cm3, so, 1 cm = 0.0864 Track cm 2 Bq · m 3 · day [20]. The relation between K theoretical and K experement in the can technique is:

K experement Track cm 2 Bq · m 3 · day = 0.0864 i K i theoretical cm E138

## 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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