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 ð Þ in the air volume of the can given by the following relation [2, 3]:

$$\mathbf{C} = \frac{A\_{Rn}}{V} = \frac{\lambda\_{Rn} N\_{Rn}}{V} \tag{9}$$

Eq. (9) becomes:

$$\mathbf{C}(t) = \mathbf{C}\_{\mathbf{t}} \left(\mathbf{1} - e^{-\lambda\_{\text{Re}}t}\right) \tag{10}$$

Where *C t*ð Þ is the radon concentration at time *t Bq* � *m*�3 ð Þ, *Cs* is the steady state (final) concentration *Bq* � *m*�3 ð Þ. 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].

$$
\rho = \frac{N\_{\text{total\\_track}}}{A\_D} \tag{11}
$$

where *ρ* is the track density expressed in Track � cm�2 ð Þ, *AD* 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 *AFOV*. Eq. (11) becomes:

$$\rho = \frac{N\_{\text{total\\_track}}}{n\text{ }A\_{FOV}} = \frac{N\_{\text{avg}}}{A\_{FOV}} \tag{12}$$

where *n* the number of fields is, Navg is the average of total tracks and AFOV 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].

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

$$\frac{d\rho}{dt} \text{ocC}(t) \tag{13}$$

or

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 *ΑRa*ð Þ 0 by steady

*ARn*ðÞ¼ *t As* 1 � *e*

*C* ¼ *ARn*

*C t*ðÞ¼ *Cs* 1 � *e*

(final) concentration *Bq* � *m*�3 ð Þ. Eq. (10) is the well-known equation which describes the buildup of the concentration of radon emanated from each sample

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 ð Þ in the

*V* ¼ *λRnNRn*

Where *C t*ð Þ is the radon concentration at time *t Bq* � *m*�3 ð Þ, *Cs* is the steady state

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

> *ρ* ¼ *Ntotal track AD*

). Since the etching track is observed and accounted by using

¼ *Navg AFOV*

). The measured track density rate recorded on the

where *n* the number of fields is, Navg is the average of total tracks and AFOV is the

SSNTD is proportional to the radon concentration during the time of exposure [7].

where *ρ* is the track density expressed in Track � cm�2 ð Þ, *AD* is the area of

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 *AFOV*. Eq. (11) becomes:

> *ρ* ¼ *Ntotal track n AFOV*

�*λRnt* (8)

*V* (9)

(11)

(12)

�*λRnt* (10)

state (final) activity of radon A*s* in Eq. (7) to become:

*Recent Techniques and Applications in Ionizing Radiation Research*

air volume of the can given by the following relation [2, 3]:

inside the exhalation container with time [1, 6].

**2.2 Track density-radon concentration relation**

Eq. (9) becomes:

detector, i.e. [1].

detector in (cm2

**74**

area of the field of view (cm2

$$\frac{d\rho}{dt} = K \text{ C(t)}\tag{14}$$

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*ð Þ¼ 0 0:

$$\rho = K \int\_0^T \mathbf{C}(t) \, dt \tag{15}$$

where *C t*ð Þ 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]:

$$\mathbf{C}\_{I} = \int\_{0}^{\mathbf{T}} \mathbf{C}(\mathbf{t}) \, \mathbf{d}\mathbf{t} \tag{16}$$

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

$$\mathbf{C}\_{I} = \frac{\rho}{\mathbf{K}} \tag{17}$$

2.We define the average radon concentration (Bq � m�3 ) by the expression of:

$$\mathbf{C}\_{\text{avg}} = \frac{1}{T} \int\_{0}^{T} \mathbf{C}(t) \, dt \tag{18}$$

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

$$\mathbf{C}\_{\text{avg}} = \frac{\rho}{K \, T} \tag{19}$$

3.The radon concentration at steady state value Cs is:

$$\mathbf{C}\_{s} = \frac{\rho}{\mathbf{K} \, T\_{27d}} \tag{20}$$

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

area of sample from which the exhalation takes place), λ is the decay constant of

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

volume of the can as a function of time t that can be given by the following relation

*N t*ðÞ¼ *EAAτ λ*

*C t*ðÞ¼ *A t*ð Þ

*C t*ðÞ¼ *EA A τ V*

*C t*ðÞ¼ *EA A λV*

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

At steady state (secular equilibrium), *dN t*ð Þ

*But τ*ð Þ¼ *h* 1

*CI* ¼ *E A*

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

*EA* ¼ *CI λ V A Teff*

*EA* ¼ *Cs λ V*

And the exhalation radon rate as a function average radon concentration is

*EA* ¼ *Cavg λ V T A Teff*

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

*EA* ¼ *ρ λ V KA Teff*

) and *τ* is the live time of radon (h). The quantity *τ* does not exist in

1 � *e*

*V* ¼ *λ N t*ð Þ

1 � *e*

1 � *e*

�*λt* (22)

*V* (23)

�*λt* (24)

*λ h*�1 (25)

�*λt* (26)

*λV Teff* (27)

*dt* ¼ 0 *and Teff* ¼ *T*, we get:

*A* (29)

(28)

(30)

(31)

), the activity concentration of radon *C*(*t*) in the air

radon (h�1

Eq. (23):

given by:

given by:

density:

**77**

*N*(*0*) = *0* is Eq. (22):

If *V* is the volume of air (m3

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

The equation becomes:

**Table 1.**

*Radon concentration in different quantities.*

The results lead to the following remarks [1]:

#### **2.3 Radon exhalation rate equation**

When radium decay in soil, the resulting atoms of radon isotopes first escapes from the mineral to air-filled pores. Being a noble gas, 222Rn can move large distances through rocks and soils. Radon can diffuse through rocks and soil, can move from one place to the other and can leak out in the atmosphere from the soil. The exhalation of radon from soil involves two mechanisms, the emanation and transport. The measurement of radon exhalation rate in soil is helpful to study radon health hazard. The passive measuring techniques "Can Technique" employing a Solid State Nuclear Track Detector (SSNTDs), a simple and efficient method to assess radon exhalation rates besides being relatively inexpensive, the technique provides quite reliable measurements. The exhalation rate is defined as the rate at which radon escapes from soil into the surrounding air. This may be measured by either per unit area or per unit mass of sample. Consider a sealed cylindrical can fitted with a source of radon and a SSNTD dosimeter fixed at the top of the can. Assume that radon, thoron and their daughters are in radioactive equilibrium in the air volume of the can. For diffusion in air, it is expected that all daughters of interest will be deposited except 216Po will be in air volume. Furthermore, 220Rn and 216Po are in homogenously distributed in the air due to their short half-lives. The 212Po and 212Bi formed by the decay of 216Po will be preferentially in homogenously deposited on the wall of the can. It is clear that the track density is registered on the detector which is related to 222Rn, as well as its plated-out daughters. The exhalation of radon from the sample surface represents the source of the number of radon atoms *N*(*t*), present in the air between the sample and SSNTD Which is directly proportional to area of surface and life time for radon. The natural decay of radon provides the only removal mechanism. The rate of change of *N*(*t*) with time is therefore governed by the following differential Eq. (21) [1–3]:

$$\frac{dN(t)}{dt} = E\_A A \tau - \lambda \, N(t) \tag{21}$$

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 *Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

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):

$$N(t) = \frac{E\_A A \tau}{\lambda} \text{ (1}-e^{-\lambda t} \text{)}\tag{22}$$

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):

$$\mathbf{C}(t) = \frac{\mathbf{A}(t)}{V} = \frac{\lambda \, N(t)}{V} \tag{23}$$

The equation becomes:

The results lead to the following remarks [1]:

concentration as shown in **Table 1**.

large exposure time.

**Integrated concentration** *CI* **(Bq** � **m**�**3** � **day)**

*Radon concentration in different quantities.*

*ρ K*

**Table 1.**

**2.3 Radon exhalation rate equation**

(*t* ffi 7*T*1*=*2) regardless of the volume of the container.

*ρ K T*

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therefore governed by the following differential Eq. (21) [1–3]:

the exhalation rate (Bq � m�2 � h�1

**76**

*dN t*ð Þ

where *N*(*t*) is the total number of radon atoms present in the can at time *t*, *E* is

*dt* ¼ *EAAτ* � *λ N t*ð Þ (21)

), *A* is cross-sectional area of the can (the surface

1.The radon concentration reaches the equilibrium state at the same time

**Average concentration Cavg (Bq** � **m**�**3**

**)**

**Steady state concentration**

**)**

**Cs (Bq** � **m**�**3**

*ρ K T*27*d*

2.There are three different quantities measured by track density, integrated radon concentration, average radon concentration and saturate radon

3.The average radon concentration equal to the saturate radon concentration at

When radium decay in soil, the resulting atoms of radon isotopes first escapes from the mineral to air-filled pores. Being a noble gas, 222Rn can move large distances through rocks and soils. Radon can diffuse through rocks and soil, can move from one place to the other and can leak out in the atmosphere from the soil. The exhalation of radon from soil involves two mechanisms, the emanation and transport. The measurement of radon exhalation rate in soil is helpful to study radon health hazard. The passive measuring techniques "Can Technique" employing a Solid State Nuclear Track Detector (SSNTDs), a simple and efficient method to assess radon exhalation rates besides being relatively inexpensive, the technique provides quite reliable measurements. The exhalation rate is defined as the rate at which radon escapes from soil into the surrounding air. This may be measured by either per unit area or per unit mass of sample. Consider a sealed cylindrical can fitted with a source of radon and a SSNTD dosimeter fixed at the top of the can. Assume that radon, thoron and their daughters are in radioactive equilibrium in the air volume of the can. For diffusion in air, it is expected that all daughters of interest will be deposited except 216Po will be in air volume. Furthermore, 220Rn and 216Po are in homogenously distributed in the air due to their short half-lives. The 212Po and 212Bi formed by the decay of 216Po will be preferentially in homogenously deposited on the wall of the can. It is clear that the track density is registered on the detector which is related to 222Rn, as well as its plated-out daughters. The exhalation of radon from the sample surface represents the source of the number of radon atoms *N*(*t*), present in the air between the sample and SSNTD Which is directly proportional to area of surface and life time for radon. The natural decay of radon provides the only removal mechanism. The rate of change of *N*(*t*) with time is

$$C(t) = \frac{E\_A A \text{ } \tau}{V} \text{ } (1 - e^{-\lambda t}) \tag{24}$$

$$\text{But } \pi(h) = \frac{1}{\lambda(h^{-1})} \tag{25}$$

$$\mathbf{C}(t) = \frac{E\_A \ A}{\lambda V} \ (\mathbf{1} - e^{-\lambda t}) \tag{26}$$

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

$$C\_I = \frac{EA}{\lambda V} T\_{\text{eff}} \tag{27}$$

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

$$E\_A = \frac{C\_I \lambda}{A} \frac{V}{T\_{\text{eff}}} \tag{28}$$

At steady state (secular equilibrium), *dN t*ð Þ *dt* ¼ 0 *and Teff* ¼ *T*, we get:

$$E\_A = \frac{C\_\text{s} \lambda^\text{ } V}{A} \tag{29}$$

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

$$E\_A = \frac{C\_{\text{avg}} \,\lambda \, V \, T}{A \, T\_{\text{eff}}} \tag{30}$$

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

$$E\_A = \frac{\rho \text{ } \lambda \text{ } V}{\text{KA } T\_{\text{eff}}} \tag{31}$$

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

$$\begin{array}{c} E\_{\mathcal{M}} = \frac{\mathcal{C}\_{I}\lambda\mathcal{V}}{\mathcal{M}^{\mathcal{T}\_{eff}}}\\ E\_{\mathcal{M}} = \frac{\mathcal{C}\_{\mathcal{S}}\mathcal{T}\lambda\mathcal{V}}{\mathcal{M}\mathcal{T}\_{eff}}\\ E\_{\mathcal{M}} = \frac{\mathcal{C}\_{avg}\lambda\mathcal{V}\mathcal{T}}{\mathcal{M}\mathcal{T}\_{eff}}\\ E\_{\mathcal{M}} = \frac{\rho\lambda\mathcal{V}}{\mathcal{M}\mathcal{T}\_{eff}} \end{array} \tag{32}$$

C*Ra* ¼ *ρ A h K MTeff*

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

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

progenies have bombarded the SSNTD films [1, 8].

*ρG*

*ρG*

*EA Bq m*�2 *h*�1 ¼ *CI λ V*

*EM Bq kg*�1 *h*�1 ¼ *CI λ V*

C*Ra Bq kg*�1 ¼ *CIV*

*The exhalation radon rate formulize.*

**Table 2.**

**79**

dividing right hand-Eq. (42) *CRn*, we obtain:

*Rn* ¼ *KRn* þ *KPo*218

*A Teff*

*M Teff*

*MTeff*

*ρG*

the CR-39 detector *ρG*

Since the unit of time in exhalation formula in hour, we should convert units of *λ*, *Teff* 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*, *Teff* , *K*, *M*, *A*,*V*Þ 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]:

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

The global track density rates, due to α-particles emitted by radon, registered on

where *ρi* are the track density for radon and its progenies on CR-39 detectors and

where *Ki* are the calibration factors for radon and its progenies. By malty and

*CPo*218 *CRn*

**Quantity** *CI C***s** *C*avg *ρ*

*Cs λ V A*

*Cs λ V M*

> *Cs V M*

from Eq. (19) can written by concentration average of radon and its progenies:

*Rn* ¼ *ρRn* þ *ρPo*218 þ *ρPo*214 (41)

*CPo*218 *CRn CRnT* (43)

> *CavgT λ V A Teff*

> *CavgT λ V M Teff*

> > *CavgT V MTeff*

*ρ λ V KA Teff*

*ρ λ V KMTeff*

*ρ V K MTeff*

*Rn* ¼ *KRnCRn T* þ *KPo*218*CPo*218*T* þ *KPo*214*CPo*214*T* (42)

þ *KPo*214

*Rn* can be writing mathematical:

(39)

ð40Þ

#### **2.4 Radium content calculation**

Since the half-life of radium is 1600 years and that of radon is 3.82 days, it is reasonable to assume that an effective equilibrium (about 98%) for radium-radon members of the decay series is reached to about three weeks or more. Once the radioactive equilibrium is established, one may use the radon alpha analysis for the determination of steady state activity concentration of radium. The activity concentration of radon begins to increase with time *t*, after the closing of the can, according to the relation Eq. (7) [1]:

$$\mathbf{A}\_{Rn}(t) = \mathbf{A}\_{Ra} \left(\mathbf{1} - e^{-\lambda\_{Rn}t}\right) \tag{33}$$

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

$$\mathbf{C}\_{\text{Rn}}(t) = \frac{\mathbf{A}\_{\text{Ra}}}{V} \left(\mathbf{1} - e^{-\lambda\_{\text{Ra}}t}\right) \tag{34}$$

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

$$\mathbf{A}\_{Ra} = \frac{C\_I V}{T\_{\text{eff}}} \tag{35}$$

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

$$\mathbf{C}\_{Ra} = \frac{C\_I V}{\mathbf{M} T\_{eff}} \tag{36}$$

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

$$\mathbf{C}\_{Ra} = \frac{\mathbf{A}\_{Ra}}{M} \tag{37}$$

By average radon concentration, we get:

$$\mathbf{C}\_{Ra} = \frac{\mathbf{C}\_{avg} T \, V}{M T\_{eff}} \tag{38}$$

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

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

By the same method we get the mass exhalation rate, where *M* is the mass of

*Recent Techniques and Applications in Ionizing Radiation Research*

Since the half-life of radium is 1600 years and that of radon is 3.82 days, it is reasonable to assume that an effective equilibrium (about 98%) for radium-radon members of the decay series is reached to about three weeks or more. Once the radioactive equilibrium is established, one may use the radon alpha analysis for the

determination of steady state activity concentration of radium. The activity concentration of radon begins to increase with time *t*, after the closing of the can,

Α*Rn*ðÞ¼ *t* Α*Ra* 1 � *e*

C*Rn*ðÞ¼ *t*

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

By average radon concentration, we get:

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

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

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

Α*Ra V* 1 � *e*

Α*Ra* ¼ *CIV Teff*

C*Ra* ¼ *CIV MTeff*

C*Ra* ¼ Α*Ra*

C*Ra* ¼ *CavgT V MTeff*

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

�*λRnt* (33)

�*λRnt* (34)

*M* (37)

ð32Þ

(35)

(36)

(38)

sample in container:

**2.4 Radium content calculation**

according to the relation Eq. (7) [1]:

tion of radon:

tracks density:

**78**

$$\mathbf{C}\_{\text{Rat}} = \frac{\rho \, A \, h}{K \, MT\_{\text{eff}}} \tag{39}$$

Since the unit of time in exhalation formula in hour, we should convert units of *λ*, *Teff* 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*, *Teff* , *K*, *M*, *A*,*V*Þ 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]:

$$\begin{pmatrix} \frac{E\_A}{E\_M} = \frac{\mathcal{M}}{A} \\ \frac{E\_A}{C\_{Ra}} = \lambda \frac{\mathcal{M}}{A} \\ \frac{E\_M}{C\_{Ra}} = \lambda \end{pmatrix} \tag{40}$$

#### **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 *ρG Rn* can be writing mathematical:

$$
\rho\_{\rm Rn}^G = \rho\_{\rm Rn} + \rho\_{\rm Po218} + \rho\_{\rm Po214} \tag{41}
$$

where *ρi* 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:

$$
\rho\_{Rn}^G = K\_{Rn} C\_{Rn} \ T + K\_{Po218} C\_{Po218} T + K\_{Po214} C\_{Po214} T \tag{42}
$$

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

$$
\rho\_{Rn}^G = \left( K\_{Rn} + K\_{Po218} \frac{C\_{Po218}}{C\_{Rn}} + K\_{Po214} \frac{C\_{Po218}}{C\_{Rn}} \right) C\_{Rn} T \tag{43}
$$

**Table 2.**

*The exhalation radon rate formulize.*

#### **Figure 2.**

*Closed-can technique (two different detectors).*

At radioactive secular equilibrium between radon and its progenies, i.e.:

$$\frac{\mathbf{C}\_{\text{Po218}}}{\mathbf{C}\_{\text{Rn}}} = \frac{\mathbf{C}\_{\text{Po214}}}{\mathbf{C}\_{\text{Rn}}} = \mathbf{1} \tag{44}$$

where:

**81**

*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.

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

Emin and Emax), this mean *ρr* > 1 [9].

**3. Indoor radon technique measurements**

Since, *Cr* is positive quantity, the CR-LR track density ratio should be *KRn* <*ρr* <*KTh* and *KTh* >*KRn.* Since, *ρCR* > *ρLR*, because CR-39 recorders all alpha particles energies, while LR-II detector recorders window energies (i.e. between

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

ð50Þ

ð51Þ

ð52Þ

Eq. (43) becomes:

$$
\rho\_{Rn}^G = K\_{Rn} \mathbf{C}\_{Rn} T \tag{45}
$$

where

$$K\_{R\text{\tiny{}^\circ C}} = K\_{R\text{\tiny{}^\circ C}} + K\_{P\text{\tiny{}^\circ 218}} + K\_{P\text{\tiny{}^\circ 214}} \tag{46}$$

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

$$
\rho\_{\text{CR}} = \rho\_{\text{Ru}}^{\text{G}} + \rho\_{\text{Th}}^{\text{G}} = K\_{\text{RC}} \mathbf{C}\_{\text{Ru}} T + K\_{\text{TC}} \mathbf{C}\_{\text{Th}} T \tag{47}
$$

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:

$$
\begin{bmatrix}
\rho\_{CR} = \begin{array}{c} K\_{RC} \ T \ C\_{Rn} + K\_{TC} \ TC\_{Th} \\ K\_{RL} = K\_{RL} \ T \ C\_{Rn} + K\_{TL} \ T \ C\_{Th}
\end{array} \tag{48}
$$

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

$$\mathcal{C}\_{Rn} = \frac{K\_{TL}\rho\_{CR} - K\_{TC}\rho\_{LR}}{K\_{RC}K\_{TL} - K\_{RL}K\_{TC}}$$

$$\mathcal{C}\_{Th} = \frac{K\_{RL}\rho\_{CR} - K\_{RC}\rho\_{LR}}{K\_{RL}K\_{TC} - K\_{RC}K\_{TL}}\tag{49}$$

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 *Cr* and calibration factors (*KL*,*KRn*,*KTh*,*KL*) ratio.

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

ð50Þ

$$\begin{array}{ccl} \mathcal{C}\_r = \left. K\_L \left( \frac{\rho\_r - \kappa\_{Rn}}{\kappa\_{Th} - \rho\_r} \right) \right| \\ \rho\_r = \frac{\rho\_{CR}}{\rho\_{LR}} \end{array} \tag{51}$$

$$\begin{bmatrix} K\_{Rn} = \frac{K\_{RC}}{K\_{RL}}\\ K\_{Th} = \frac{K\_{TC}}{K\_{TL}}\\ K\_L = \frac{K\_{RL}}{K\_{TL}} \end{bmatrix} \tag{52}$$

where:

At radioactive secular equilibrium between radon and its progenies, i.e.:

¼ *CPo*214 *CRn*

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:

This are tow algebraic liner equations by tow variables *CRn* and *CTh*, the general

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 *Cr* and calibration factors

¼ 1 (44)

*Rn* ¼ *KRnCCRnT* (45)

*KRnC* ¼ *KRn* þ *KPo*218 þ *KPo*214 (46)

*Th* for thoron on CR-39 detector. The total track

*Th* ¼ *KRCCRnT* þ *KTCCThT* (47)

ð48Þ

ð49Þ

*CPo*218 *CRn*

*Recent Techniques and Applications in Ionizing Radiation Research*

*ρG*

Eq. (43) becomes:

*Closed-can technique (two different detectors).*

By same procedure, we find *ρG*

density *ρCR* for radon and thoron become:

*ρCR* ¼ *ρG*

*Rn* þ *ρG*

where

**Figure 2.**

solution for this are:

(*KL*,*KRn*,*KTh*,*KL*) ratio.

**80**

*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, *Cr* is positive quantity, the CR-LR track density ratio should be *KRn* <*ρr* <*KTh* and *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

**Figure 3.** *Twin cups dosimeter.*

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]:

$$
\rho\_S = K\_{\rm RS} \, T \, \mathbf{C}\_{\rm Rn} \tag{53}
$$

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 ρB/ρS between bare (*ρB*) and in can (*ρS*) detector, respectively. There are three steps for

1.The potential alpha energy concentration of any mixture of (short-live) radon

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

The potential alpha energy concentration (*PAEC*) of any mixture of (short-live)

. This unit is related to the SI units J and m3 according to 1 J � m�3 = 6.24 �

*PAECTotal* ¼ *PAEC*1*N*1 þ *PAEC*2*N*2 þ *PAEC*3*N*3 þ *PAEC*4*N*4 (57)

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

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

*CPAET* ¼ *PAET*

*V* ¼ *λiNi*

*f* 2 þ

*PAE*3 *λ*3

, 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

*C*1 ¼ *C*2 ¼ *C*3 ¼ *C*4 ¼ *C*0 ¼ *CRn* ¼ 3*:*7 kBq*:*m�3 ¼ 3*:*7*Bq* � *l*

*f* 3 þ

*C*0 (60)

*PAE*4 *λ*4

*Ci* ¼ *Ai*

*PAE*2 *λ*2

*V* (58)

*V* (59)

*f* 4

ð61Þ

�1 (62)

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

�1 [12]. To express mathematically, we let:

calculating the equilibrium factor [1, 11, 13]:

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

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

*CPAET* ¼ *PAE*1

concentration of 3.7 kBq � m�3 [12].

Eq. (60) become as:

*λ*1

*f* 1 þ

or thoron daughters in air.

2.The working level (*WL*).

and can mode.

MeV � l

109 Mev � l

defined as:

where *fi*

**83**

�1

$$
\rho\_F = K\_{RF} \ T \ C\_{Rn} + K\_{TF} \ T \ C\_{Th} \tag{54}
$$

The general solution for this are:

$$\mathbf{C}\_{Rn} = \frac{\rho\_{\rm S}}{K\_{\rm RS}T} \tag{55}$$

$$\mathbf{C}\_{Th} = \frac{\rho\_{\rm S}}{K\_{TF}T} \left(\frac{\rho\_{\rm F}}{\rho\_{\rm S}} - \frac{\mathbf{K}\_{RF}}{\mathbf{K}\_{RS}}\right) \tag{56}$$

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, *CTh* is positive quantity, *ρF ρS* track density ratio should be *ρF ρS* > *KRF KRS:*

#### **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

#### *Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

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 ρB/ρS between bare (*ρB*) and in can (*ρS*) detector, respectively. There are three steps for calculating the equilibrium factor [1, 11, 13]:

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,

*CRn* ¼ *ρS*

*ρF ρS*

� *KRF KRS*

track density ratio should be *ρF*

*KTFT*

*CTh* ¼ *ρS*

*ρS*

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

*ρS* ¼ *KRS T CRn* (53)

*KRST* (55)

*ρS* > *KRF KRS:* (56)

*ρF* ¼ *KRF T CRn* þ *KTF T CTh* (54)

registered on the SSNTDs can write mathematical [11]:

*Recent Techniques and Applications in Ionizing Radiation Research*

The general solution for this are:

*ρS* = Tracks density for sponge cup. *ρF* = Tracks density for filter piper cup.

Since, *CTh* is positive quantity, *ρF*

**4. Equilibrium factor**

*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.

where:

**82**

**Figure 3.**

*Twin cups dosimeter.*

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:

$$PAEC\_{Total} = PAEC\_1N\_1 + PAEC\_2N\_2 + PAEC\_3N\_3 + PAEC\_4N\_4 \tag{57}$$

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:

$$\mathbf{C}\_{PAET} = \frac{\mathbf{PAET}}{V} \tag{58}$$

$$\mathbf{C}\_{i} = \frac{\mathbf{A}\_{i}}{V} = \frac{\lambda\_{i} N\_{i}}{V} \tag{59}$$

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

$$\mathbf{C}\_{PAET} = \left[\frac{\text{PAE}\_1}{\lambda\_1}f\_1 + \frac{\text{PAE}\_2}{\lambda\_2}f\_2 + \frac{\text{PAE}\_3}{\lambda\_3}f\_3 + \frac{\text{PAE}\_4}{\lambda\_4}f\_4\right] \mathbf{C}\_0 \tag{60}$$

$$\begin{aligned} f\_l &= \frac{C\_l}{C\_0} \\ C\_0 &= C\_{\mathcal{R}n} \end{aligned} \tag{61}$$

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].

$$\mathbf{C}\_1 = \mathbf{C}\_2 = \mathbf{C}\_3 = \mathbf{C}\_4 = \mathbf{C}\_0 = \mathbf{C}\_{\text{Rt}} = \mathbf{3.7} \text{ kBq.} \text{m}^{-3} = \mathbf{3.7} \text{Bq} \cdot \text{l}^{-1} \tag{62}$$

Eq. (60) become as:

$$\mathbf{C\_{PAECT}} = \mathbf{W}\mathbf{L} = \frac{\mathbf{3.7}\,\text{PE}\_1}{\lambda\_1} + \frac{\mathbf{3.7}\,\text{PE}\_2}{\lambda\_2} + \frac{\mathbf{3.7}\,\text{PE}\_3}{\lambda\_3} + \frac{\mathbf{3.7}\,\text{PE}\_4}{\lambda\_4} = \mathbf{1.3} \ast \mathbf{10^5}\,\text{MeV}\,\text{l}^{-1} \tag{63}$$

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

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 :

$$C\_{PET} = \left[\frac{\frac{3.7PE\_1}{\lambda\_1}}{1.3 \ast 10^5} f\_1 + \frac{\frac{3.7PE\_2}{\lambda\_2}}{1.3 \ast 10^5} f\_2 + \frac{\frac{3.7PE\_3}{\lambda\_3}}{1.3 \ast 10^5} f\_3 + \frac{\frac{3.7PE\_4}{\lambda\_4}}{1.3 \ast 10^5} f\_4\right] \frac{1.3 \ast 10^5}{3.7} C\_{Ru} \tag{64}$$

$$\mathbf{C\_{PET}} = \left[ \mathbf{0.11} \, f\_1 + \mathbf{0.51} \, f\_2 + \mathbf{0.38} \, f\_3 \right] \frac{\mathbf{1.3} \ast \mathbf{10^5}}{\mathbf{3.7}} \mathbf{C\_{Rn}} \tag{65}$$

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

$$C\_{PET}(\text{WL}) = \frac{F\_{Rn} \ C\_{Rn} \left( Bq \ l^{-1} \right)}{3.7} \tag{66}$$

or

$$C\_{PET}(m\text{WL}) = \frac{F\_{Rn}C\_{Rn}\,\,(Bq\,\,m^{-3})}{3.7} \tag{67}$$

where

$$F\_{Rn} = 0.11f\_1 + 0.51f\_2 + 0.38f\_3 \tag{68}$$

*FRn* and *FTn* 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

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 *λi*�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, *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]:

*dt* ¼ *λi*�1*Ni*�1 � Λ*iNi* ð Þ i ¼ 1 … 4 (71)

Λ*i* ¼ *V* þ *Ai* þ *Wi* þ *λi* (72)

*λi*�1*Ni*�1 ¼ Λ*iNi* (73)

rate equations for serial radioactive decay chain (Batman equations):

*dNi*

where

**85**

**Daughters of thoron**

**Table 4.**

**Table 5.**

**Daughters of thoron**

**α-energy (Mev)**

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

> **α-energy (Mev)**

*Numerical calculations of Eq. (63) for thoron.*

**Half-life (s)**

**Half-life (s)**

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

**Ultimate alpha energy (Mev)**

**Ultimate alpha energy (Mev)**

Po-216 6.78 0.15 14.5952 3.16E+00 0.275

*Numerical calculations of thoron concentration as corresponding to the total energy 1.3* � *105 Mev* � *l*

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

**Total energy (Mev)**

4.73E+05

**Total energy (Mev)**

1.3E+05

**CTn (Bq** � **l** �**1 )**

> �*1 .*

**Fraction energy**

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**.

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

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

$$\mathcal{C}\_{\rm PET}(m\mathcal{W}\mathcal{L}) = \frac{F\_{\rm Tn}\mathcal{C}\_{\rm Tn}(Bq.m^{-3})}{0.275} \tag{69}$$

where

$$F\_{\rm Th} = 0.91f\_2 + 0.09f\_3 \tag{70}$$

**Table 3.** *Numerical calculations of Eq. (63).*

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

**Table 4.**

*CPAECT* ¼ *WL* ¼ 3*:*7 *PE*1

*CPET* ¼

or

where

in **Table 4**.

where

**Daughters of radon**

**Table 3.**

**84**

*Numerical calculations of Eq. (63).*

as shown in **Table 5**.

*λ*1

One (*WL*) equal to 1.3 � 105 Mev � l

and multi and divide by 1.3 � 105 Mev � l

3*:*7*PE*1 *λ*1 1*:*3 ∗ 105 *f* 1 þ

þ

*Recent Techniques and Applications in Ionizing Radiation Research*

3*:*7 *PE*2 *λ*2

3*:*7*PE*2 *λ*2 1*:*3 ∗ 105 *f* 2 þ

*CPET* ¼ 0*:*11 *f* 1 þ 0*:*51 *f* 2 þ 0*:*38 *f* 3

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

*CPET*ð Þ¼ *WL*

*CPET*ð Þ¼ *mWL*

energy of alpha its daughters equal to 1.3 � 105 Mev � l

**α-energy (Mev)**

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

þ

3*:*7 *PE*3 *λ*3

potential alpha energy by fraction energy, Eq. (63) multi and divide by 3.7 Bq � l

" #

�1 :

3*:*7*PE*3 *λ*3 1*:*3 ∗ 105 *f* 3 þ

� � 1*:*3 ∗ 105

For Thoron daughters we find the concentration of thoron which give potential

By the same setup, we drive Eq. (60) for thoron and numerical calculations of it,

*CPET*ð Þ¼ *mWL FTnCTn Bq:m*�3 ð Þ

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

**Ultimate alpha energy (Mev)**

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

**Half-life (s)**

*FRn CRn Bq l*�1 � �

*FRnCRn Bq m*�3 ð Þ

*FRn* ¼ 0*:*11 *f* 1 þ 0*:*51 *f* 2 þ 0*:*38 *f* 3 (68)

*FTn* ¼ 0*:*91 *f* 2 þ 0*:*09 *f* 3 (70)

*CPAECT* ¼ 1.3E+05

þ

3*:*7 *PE*4 *λ*4

�1 of air. To obtain the concentration

3*:*7*PE*4 *λ*4 1*:*3 ∗ 105 *f* 4

3*:*7

3*:*7 (66)

3*:*7 (67)

�1 is 0.275 Bq � l

0*:*275 (69)

**Total energy (Mev)**

�1

, as shown

**Fraction energy**

¼ 1*:*3 ∗ 105 *Mev l*�1 (63)

1*:*3 ∗ 105 3*:*7

*CRn* (65)

�1

*CRn*

(64)

*Numerical calculations of thoron concentration as corresponding to the total energy 1.3* � *105 Mev* � *l* �*1 .*

#### **Table 5.**

*Numerical calculations of Eq. (63) for thoron.*

*FRn* and *FTn* 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):

$$\frac{d\mathbf{N}\_i}{dt} = \lambda\_{i-1}\mathbf{N}\_{i-1} - \Lambda\_i\mathbf{N}\_i \text{ (i = 1 \dots 4)}\tag{71}$$

where

$$
\Lambda\_i = V + A\_i + W\_i + \lambda\_i \tag{72}
$$

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 *λi*�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, *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]:

$$
\lambda\_{i-1} \mathbf{N}\_{i-1} = \Lambda\_i \mathbf{N}\_i \tag{73}
$$

*Recent Techniques and Applications in Ionizing Radiation Research*

or

$$\frac{C\_i}{C\_{i-1}} = \frac{\lambda\_i}{\Lambda\_i} = d\_i \tag{74}$$

ð86Þ

ð89Þ

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

ð Þ *λ*1 þ *V* 0*:*11 þ 0*:*51 *λ*2

*FTn* ¼ *λ*1*λ*2

*λ*1 ð Þ *λ*1 þ *V*

*λ*2 ð Þ *λ*2 þ *V*

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).

ð Þ *λ*1 þ *V* ð Þ *λ*2 þ *V* 0*:*91 þ 0*:*09 *λ*3

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

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

The diffusion of radon is a process determined by radon gas concentration gradient across the radon gas sources (rocks, soils, building materials, and other different materials) and the surrounding air. The knowledge of radon diffusion coefficient makes it possible to determine other features of a material related to radon. The diffusion of radon through the porous medium can be described by Fick's second laws [1, 14, 15]. The basic concept in measuring radon concentration involves observations of radon diffusion through the porous medium being studied. This medium separates two chambers that have different radon concentrations. In such configurations, radon diffusion comes from one chamber to another, primarily in one direction only. The standard method for the measurement of radon diffusion coefficient beads on the evaluation of radon flux through tested material exposed to high radon concentration and placed between two containers by measuring in both sides of material. For this, one of the containers is connected to radon source (radium source) and the other where the radon concentration is periodically evaluated as shown in **Figure 4**. It is assumed that radon is well mixed in both containers.

ð Þ *λ*2 þ *V* þ 0*:*38 *λ*2*λ*3

(90)

ð Þ *λ*2 þ *V* ð Þ *λ*3 þ *V*

ð Þ *λ*3 þ *V* (91)

*λ*3

*V*3 þ *a*2*V*2 þ *a*1 *V* þ *a*0 ¼ 0 (88)

ð Þ *λ*3 þ *V* ¼ *KSBρBS* � 1 (87)

*λ*1 ð Þ *λ*1 þ *V* þ

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

After some setup, we obtain:

where

*FRn* ¼ *λ*1

**87**

combined Eq. (90) and Eq. (91).

From Eq. (74), we obtain:

$$\begin{aligned} f\_1 &= \frac{c\_1}{c\_0} = d\_1\\ f\_2 &= \frac{C\_2}{C\_0} = d\_1 d\_2\\ f\_3 &= \frac{C\_3}{C\_0} = d\_1 d\_2 d\_3\\ f\_4 &= \frac{c\_4}{c\_0} = d\_1 d\_2 d\_3 \end{aligned} \tag{75}$$

where *C*4 *C*3 ¼ *d*4 ¼ 1 since *C*3 ¼ *C*4For *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:

$$
\Lambda\_i = V + \lambda\_i \tag{76}
$$

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:

$$
\rho\_{\mathbb{S}} = \rho\_0 + \rho\_1 + \rho\_4 \tag{77}
$$

$$
\rho\_S = K\_0 C\_0 T + K\_1 C\_1 T + K\_4 C\_4 T \tag{78}
$$

$$
\rho\_S = \left(K\_0 + K\_1 f\_1 + K\_4 f\_4\right) \mathbf{C}\_0 T \tag{79}
$$

In can mode *f* 1 ¼ *f* 2 ¼ *f* 3 ¼ *f* 4 ¼ 1, Eq. (79) become as:

$$
\rho\_S = K\_S C\_0 T \tag{80}
$$

where

$$K\_t = K\_0 + K\_1 + K\_4 \tag{81}$$

In bare mode (in absence thoron)

$$
\rho\_B = \rho\_0 + \rho\_1 + \rho\_4 \tag{82}
$$

$$
\rho\_B = (\overline{K}\_0 + \overline{K}\_1 f\_1 + \overline{K}\_4 f\_4) \overline{C}\_0 T \tag{83}
$$

In bare mode *f* 1 6¼ *f* 2 6¼ *f* 3 6¼ *f* 4 because no equilibrium between the radon and its progenies, but *K*0 ¼ *K*1 ¼ *K*4 ¼ *KB*, therefore Eq. (83) become as:

$$
\rho\_B = K\_B (1 + f\_1 + f\_4) \overline{\mathbf{C}}\_0 T \tag{84}
$$

By dividing Eq. (84) on Eq. (80) when *C*0 ¼ *C*0, we obtain:

$$\mathbf{1} + f\_1 + f\_4 = K\_{\text{SB}} \rho\_{\text{BS}} \tag{85}$$

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

$$\begin{aligned} \rho\_{BS} &= \frac{\rho\_B}{\rho\_S} \\ \mathbf{K}\_{SB} &= \frac{K\_S}{\kappa\_B} \end{aligned} \qquad\tag{86}$$

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

$$\frac{\lambda\_1}{\left(\lambda\_1 + V\right)} + \frac{\lambda\_1}{\left(\lambda\_1 + V\right)} \frac{\lambda\_2}{\left(\lambda\_2 + V\right)} \frac{\lambda\_3}{\left(\lambda\_3 + V\right)} = K\_{SB} \rho\_{BS} - 1 \tag{87}$$

After some setup, we obtain:

$$V^3 + a\_2V^2 + a\_1\ V + a\_0 = 0\tag{88}$$

where

or

where *C*4

In can mode:

where

**86**

From Eq. (74), we obtain:

by track density measurements, so:

In bare mode (in absence thoron)

*Ci Ci*�1

*C*3 ¼ *d*4 ¼ 1 since *C*3 ¼ *C*4For *λ*3 ≪ *λ*4

*Recent Techniques and Applications in Ionizing Radiation Research*

relates the concentration of radon and its daughters as:

¼ *λi* Λ*i*

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

The track density of both bare mode and can mode (with sponge filter) detector

*ρS* ¼ *K*0 þ *K*1 *f* 1 þ *K*4 *f* 4

*ρB* ¼ *K*0 þ *K*1 *f* 1 þ *K*4 *f* 4

*ρB* ¼ *KB* 1 þ *f* 1 þ *f* 4

its progenies, but *K*0 ¼ *K*1 ¼ *K*4 ¼ *KB*, therefore Eq. (83) become as:

By dividing Eq. (84) on Eq. (80) when *C*0 ¼ *C*0, we obtain:

In bare mode *f* 1 6¼ *f* 2 6¼ *f* 3 6¼ *f* 4 because no equilibrium between the radon and

In can mode *f* 1 ¼ *f* 2 ¼ *f* 3 ¼ *f* 4 ¼ 1, Eq. (79) become as:

¼ *di* (74)

Λ*i* ¼ *V* þ *λi* (76)

*ρS* ¼ *ρ*0 þ *ρ*1 þ *ρ*4 (77)

*C*0*T* (79)

*ρS* ¼ *KSC*0*T* (80)

*Ks* ¼ *K*0 þ *K*1 þ *K*4 (81)

*ρB* ¼ *ρ*0 þ *ρ*1 þ *ρ*4 (82)

*C*0*T* (83)

*C*0*T* (84)

1 þ *f* 1 þ *f* 4 ¼ *KSBρBS* (85)

*ρS* ¼ *K*0*C*0*T* þ *K*1*C*1*T* þ *K*4*C*4*T* (78)

ð75Þ

$$\begin{array}{l} a\_2 = (1+\mathcal{B})\lambda\_1 + \lambda\_2 + \lambda\_3\\ a\_1 = (1+\mathcal{B})(\lambda\_2 + \lambda\_3)\lambda\_1 + \lambda\_2\lambda\_3\\ a\_0 = (1+2\mathcal{B})\lambda\_1\lambda\_2\lambda\_3\\ \mathcal{B} = \frac{1}{1 - K\_{SB}\rho\_{BS}} \end{array} \tag{89}$$

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).

$$F\_{R\pi} = \frac{\lambda\_1}{(\lambda\_1 + V)} \left( 0.11 + 0.51 \frac{\lambda\_2}{(\lambda\_2 + V)} + 0.38 \frac{\lambda\_2 \lambda\_3}{(\lambda\_2 + V)(\lambda\_3 + V)} \right) \tag{90}$$

$$F\_{\rm T\pi} = \frac{\lambda\_1 \lambda\_2}{(\lambda\_1 + V)(\lambda\_2 + V)} \left( 0.91 + 0.09 \,\frac{\lambda\_3}{(\lambda\_3 + V)} \right) \tag{91}$$

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 diffusion of radon is a process determined by radon gas concentration gradient across the radon gas sources (rocks, soils, building materials, and other different materials) and the surrounding air. The knowledge of radon diffusion coefficient makes it possible to determine other features of a material related to radon. The diffusion of radon through the porous medium can be described by Fick's second laws [1, 14, 15]. The basic concept in measuring radon concentration involves observations of radon diffusion through the porous medium being studied. This medium separates two chambers that have different radon concentrations. In such configurations, radon diffusion comes from one chamber to another, primarily in one direction only. The standard method for the measurement of radon diffusion coefficient beads on the evaluation of radon flux through tested material exposed to high radon concentration and placed between two containers by measuring in both sides of material. For this, one of the containers is connected to radon source (radium source) and the other where the radon concentration is periodically evaluated as shown in **Figure 4**. It is assumed that radon is well mixed in both containers.

**Figure 4.** *Diffusion radon chambers.*

The determination of the radon diffusion coefficient with this method is normally performed under steady state conditions.

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 timedependent one-dimensional differential equation (no convection) describing the radon activity concentration C0 is given by Fick' s second law:

$$\frac{\partial \mathcal{C}\_0(\mathbf{z}, t)}{\partial t} = D\_\varepsilon \frac{\partial^2 \mathcal{C}\_0(\mathbf{z}, t)}{\partial \mathbf{z}^2} - \lambda \, \mathcal{C}\_0(\mathbf{z}, t) + P \tag{92}$$

where *C*0ð Þ *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):

$$D\_{\varepsilon} = \frac{D}{\varepsilon} \tag{93}$$

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

we needed:

where

We get two equations:

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

By using the relations:

sinh *d*�*z l* � �

sinh *d l* � � *C*1 þ

s first law:

*l* � �

*dz* ¼ � cosh *d*�*z*

*l*sinh *d l* � � *C*1 þ

*C*0ð Þ¼ *z*

From Fick'

We get:

*dC*0ð Þ*z*

**89**

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,

*C*1 ¼ *A* þ *B* þ

*C*2 ¼ *A eβ* þ *B e*�*β* þ

*β* ¼ *d*

1

*eβ* � *e*�*β C*2 þ

sinh *d*�*z l*

*E* ¼ *λ V C*2

*dC*0ð Þ*z dz*

� � � � *z*¼*d*

cosh *z α*

*E* ¼ �*De*

cosh *z l* � �

*l*sinh *d l* � �*C*2 þ

*eβ* � *e*�*β C*2 � 1 � *e*�*β*

*eβ* � *e*�*β*

1 � *eβ eβ* � *e*�*β*

� � þ sinh *z*

! *P*

sinh *d l*

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

*A* ¼ �*e*�*β*

*B* ¼ *eβ*

*eβ* � *e*�*β C*1 þ

And mathematical steps, we get the general solution:

sinh *z l* � �

sinh *d l* � �*C*2 þ

From steady state of exhalation radon Eq. (29), we get:

*eβ* � *e*�*β C*1 � 1

*P*

*P*

*λ* (99)

*l* (101)

*P*

*P*

*l* � �

� � � 1

*A* (106)

� � � cosh *d*�*z*

*l*sinh *d l* � � ! *P*

*α* � �

*λ* (102)

*λ* (103)

ð104Þ

*λ* (105)

(107)

*λ* (108)

*λ* (100)

ð98Þ

$$P = \frac{f \text{ } \lambda \text{ C}\_{\text{Ra}} \text{ } \rho \text{ }}{\varepsilon} \tag{94}$$

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

In steady state *∂C*0ð Þ *z*, *t ∂t* ¼ 0 and Eq. (92) becomes:

$$\frac{d^2\mathcal{C}\_0(z)}{dz^2} - \frac{\lambda}{D\_\epsilon}\mathcal{C}\_0(z) + \frac{P}{D\_\epsilon} = \mathbf{0} \tag{95}$$

The general solution is:

$$\mathbf{C}\_{0}(x) = A \, e^{x/l} + B \, e^{-x/l} + \frac{P}{\lambda} \tag{96}$$

$$d = \sqrt{\frac{D\_e}{\lambda}}\tag{97}$$

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

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:

$$\begin{cases} \mathcal{C}\_0(\mathbf{z} = 0) = \mathcal{C}\_1 \\ \mathcal{C}\_0(\mathbf{z} = d) = \mathcal{C}\_2 \end{cases} \right\} \tag{98}$$

We get two equations:

$$\mathbf{C}\_1 = \mathbf{A} + \mathbf{B} + \frac{P}{\lambda} \tag{99}$$

$$\mathbf{C}\_2 = \mathbf{A} \,\, \mathbf{e}^{\beta} + \mathbf{B} \,\, \mathbf{e}^{-\beta} + \frac{P}{\lambda} \,\, \tag{100}$$

where

The determination of the radon diffusion coefficient with this method is normally

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 timedependent one-dimensional differential equation (no convection) describing the

*C*0ð Þ *z*, *t*

*De* is the effective diffusion coefficient *λ* is the radon decay constant, *ε* is the prostiy

*De* ¼ *D*

*P* ¼ *f λ CRa ρ*

s second law:

/h, gas flux expressed per unit area of material as a whole),

*∂z*2 � *λ C*0ð Þþ *z*, *t P* (92)

*ε* (93)

*ε* (94)

¼ 0 (95)

*λ* (96)

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

), *D* is the bulk

) and *CRa* is the

(97)

performed under steady state conditions.

*Recent Techniques and Applications in Ionizing Radiation Research*

radon activity concentration C0 is given by Fick'

diffusion coefficient (m2

**Figure 4.**

*Diffusion radon chambers.*

radium concentration (Bq/kg). In steady state *∂C*0ð Þ *z*, *t*

The general solution is:

**88**

*∂C*0ð Þ *z*, *t*

*∂t* ¼ *De*

and *P* is the production rate of radon in the pore air (Bq/m3 � h):

*∂*2

where *C*0ð Þ *z*, *t* is the radon concentration within the pores (Bq/m3

where, *f* is the emanation fraction, *ρ* is the bulk density (kg/m3

*∂t* ¼ 0 and Eq.  (99) and (100) and some mathematical step, we get:

$$A = \frac{-e^{-\beta}}{e^{\beta} - e^{-\beta}} \mathbf{C}\_1 + \frac{\mathbf{1}}{e^{\beta} - e^{-\beta}} \mathbf{C}\_2 - \frac{\mathbf{1} - e^{-\beta}}{e^{\beta} - e^{-\beta}} \frac{P}{\lambda} \tag{102}$$

$$B = \frac{e^{\beta}}{e^{\beta} - e^{-\beta}} \mathbf{C}\_1 - \frac{\mathbf{1}}{e^{\beta} - e^{-\beta}} \mathbf{C}\_2 + \frac{\mathbf{1} - e^{\beta}}{e^{\beta} - e^{-\beta}} \frac{P}{\lambda} \tag{103}$$

By using the relations:

$$\begin{aligned} e^{\beta} + e^{-\beta} &= 2 \cosh \beta \\ e^{\beta} - e^{-\beta} &= 2 \sinh \beta \end{aligned} \qquad \tag{104}$$

And mathematical steps, we get the general solution:

$$\mathbf{C}\_{0}(\mathbf{z}) = \frac{\sinh\left(\frac{d-\mathbf{z}}{l}\right)}{\sinh\left(\frac{d}{l}\right)} \mathbf{C}\_{1} + \frac{\sinh\left(\frac{\mathbf{z}}{l}\right)}{\sinh\left(\frac{d}{l}\right)} \mathbf{C}\_{2} + \left(\frac{\sinh\left(\frac{d-\mathbf{z}}{l}\right) + \sinh\left(\frac{\mathbf{z}}{l}\right)}{\sinh\left(\frac{d}{l}\right)} - \mathbf{1}\right) \frac{P}{\lambda} \tag{105}$$

From steady state of exhalation radon Eq. (29), we get:

$$E = \frac{\lambda \text{ V C}\_2}{A} \tag{106}$$

From Fick' s first law:

$$E = -D\_\epsilon \frac{dC\_0(z)}{dz}\bigg|\_{z=d} \tag{107}$$

We get:

$$\frac{d\mathbf{C}\_{0}(\mathbf{z})}{dz} = \frac{-\cosh\left(\frac{d-\mathbf{z}}{l}\right)}{l\sinh\left(\frac{d}{l}\right)}\mathbf{C}\_{1} + \frac{\cosh\left(\frac{\mathbf{z}}{l}\right)}{l\sinh\left(\frac{d}{l}\right)}\mathbf{C}\_{2} + \left(\frac{\cosh\left(\frac{\mathbf{z}}{a}\right) - \cosh\left(\frac{d-\mathbf{z}}{a}\right)}{l\sinh\left(\frac{d}{l}\right)}\right)\frac{P}{\lambda} \tag{108}$$

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$$\left. \frac{d\mathbf{C}\_0(z)}{dz} \right|\_{z=d} = \frac{-\mathbf{1}}{l \sinh\left(\frac{d}{l}\right)} \mathbf{C}\_1 + \frac{\cosh\left(\frac{d}{l}\right)}{l \sinh\left(\frac{d}{l}\right)} \mathbf{C}\_2 + \left(\frac{\cosh\left(\frac{d}{a}\right) - \mathbf{1}}{l \sinh\left(\frac{d}{l}\right)}\right) \frac{P}{\lambda} \tag{109}$$

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

$$E = \frac{-D\_\epsilon}{l \sinh\left(\frac{d}{l}\right)} \left\{-\mathbf{C}\_1 + \cosh\left(\frac{d}{l}\right)\mathbf{C}\_2 - \left(\cosh\left(\frac{d}{l}\right) - \mathbf{1}\right)\frac{P}{\lambda}\right\} \tag{110}$$

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

$$C\_2 = \frac{C\_1}{\cosh\left(\frac{d}{l}\right) + \frac{l l V\_2}{D\_l A} \sinh\left(\frac{d}{l}\right)} + \frac{1 - \cosh\left(\frac{d}{l}\right)}{\cosh\left(\frac{d}{l}\right) + \frac{l l V\_2}{D\_l A} \sinh\left(\frac{d}{l}\right)} \frac{P}{\lambda} \tag{111}$$

Since,

$$\begin{cases} V\_2 = A \, h \\ l^2 = \frac{D\_e}{\lambda} \\ \frac{\lambda \nu\_2}{D\_0 A} = \frac{h}{l} \end{cases} \tag{112}$$

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

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

*D*1

*D*2

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

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

*x=l*1 þ *B*1*e*

*x=l*2 þ *B*2*e*

*C*1ð Þ¼ *x* 0 (118)

*C*2ð Þ¼ *x* 0 (119)

�*x=l*1 (120)

�*x=l*2 (121)

ð122Þ

ð123Þ

radon diffusion in x-direction **Figure 5** [1, 14, 16].

*d*2 *C*1ð Þ *x dx*2 � *λ*

*d*2 *C*2ð Þ *x dx*2 � *λ*

*C*1ð Þ¼ *x A*1*e*

*C*2ð Þ¼ *x A*2*e*

the lack of a source of radium) respectively:

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

The general solutions are

**Figure 5.** *Diffusion radon cup.*

porous filter and air respectively.

**91**

The solutions of these equations are:

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

$$C\_2 = \frac{C\_1}{\cosh\left(\frac{d}{l}\right) + \frac{h}{l}\sinh\left(\frac{d}{l}\right)} + \frac{1 - \cosh\left(\frac{d}{l}\right)}{\cosh\left(\frac{d}{l}\right) + \frac{h}{l}\sinh\left(\frac{d}{l}\right)} \frac{P}{\lambda} \tag{113}$$

Rewrite Eq. (113):

$$\left(1 - \frac{P}{\lambda \, C\_2}\right) \cosh\left(\frac{d}{l}\right) + \frac{h}{l} \sinh\left(\frac{d}{l}\right) - \left(\frac{C\_1}{C\_2} - \frac{P}{\lambda \, C\_2}\right) = 0\tag{114}$$

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

$$\left(\mathbf{1} - \frac{P}{\lambda \,\mathbf{C}\_2}\right) \cosh \beta + \frac{h}{d} \beta \sinh \beta - \left(\frac{\mathbf{C}\_1}{\mathbf{C}\_2} - \frac{P}{\lambda \,\mathbf{C}\_2}\right) = \mathbf{0} \tag{115}$$

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:

$$\begin{aligned} \sinh \beta &= \frac{\,^{\beta \cdot \beta \cdots \beta} \,^{-\beta}}{\,^{2} \cosh \beta} = \frac{1 + \beta - 1 + \beta}{2} = \beta \\\cosh \beta &= \frac{\,^{\alpha \cdot \beta \cdots \alpha \cdot \beta}}{2} = \frac{1 + \beta + 1 - \beta}{2} = 1 \end{aligned} \tag{116}$$

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

$$D\_{\epsilon} = \frac{\lambda \, d \, h}{\frac{\epsilon}{C\_2} - 1} \tag{117}$$

#### **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

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

**Figure 5.** *Diffusion radon cup.*

*dC*0ð Þ*z dz*

� � � � *z*¼*d*

*E* ¼ �*De l* sinh *d l*

Rewrite Eq. (113):

1 � *P λ C*2 � �

**6. Transmission factor**

**90**

Since,

¼ �1 *l*sinh *d l* � �*C*1 þ

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

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

*C*2 ¼ *C*1 cosh *d l* � � þ *h*

> cosh *d l* � � þ *h*

1 � *P λ C*2 � �

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

cosh *β* þ

be numerically calculated by using the Newton-Raphson method.

to evolution radon diffusion coefficient by simple relation as:

*C*2 ¼ *C*1 cosh *d l* � � þ *λlV*2

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

� � �*C*1 þ cosh *d*

*Recent Techniques and Applications in Ionizing Radiation Research*

*DeA* sinh *d*

cosh *d l* � � cosh *d α* � � � 1 *l*sinh *d l* � �

> *l* � �

> > *l* � �

*DeA* sinh *d*

1 � cosh *d*

� *P λ C*2 � �

� *P λ C*2 � �

*C*2 � 1 (117)

cosh *d l* � � þ *h*

� *C*1 *C*2

*C*2

*d β* sinh *β* � *C*1

Eq. (115) is nonlinear equation; therefore, the radon diffusion coefficient *D* can

IF *β* is small, and by using set relation in Eq. (116), we can make approximation

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

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

*De* ¼ *λ d h C*

*l* � �

*l* sinh *d l* � � *P*

1 � cosh *d*

� � *P*

� 1

*λ*

*l* � � *P*

*C*2 � cosh *d*

� �

cosh *d l* � � þ *λlV*2

!

*P*

*λ* (109)

*λ* (111)

*λ* (113)

¼ 0 (114)

¼ 0 (115)

ð116Þ

(110)

ð112Þ

*l*sinh *d l* � �*C*2 þ

*l* � �

*l* � � þ

*l* sinh *d l* � � þ

> *l* sinh *d l* � �

> > *h*

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:

$$\frac{d^2\mathbf{C}\_1(\mathbf{x})}{d\mathbf{x}^2} - \frac{\lambda}{D\_1}\mathbf{C}\_1(\mathbf{x}) = \mathbf{0} \tag{118}$$

$$\frac{d^2\mathbf{C}\_2(\mathbf{x})}{d\mathbf{x}^2} - \frac{\lambda}{D\_2}\mathbf{C}\_2(\mathbf{x}) = \mathbf{0} \tag{119}$$

The general solutions are

$$\mathbf{C}\_{1}(\mathbf{x}) = A\_{1}\mathbf{e}^{\mathbf{x}/l\_{1}} + B\_{1}\mathbf{e}^{-\mathbf{x}/l\_{1}} \tag{120}$$

$$\mathbf{C}\_{2}(\mathbf{x}) = \mathbf{A}\_{2}\mathbf{e}^{\mathbf{x}/l\_{2}} + \mathbf{B}\_{2}\mathbf{e}^{-\mathbf{x}/l\_{2}} \tag{121}$$

$$l\_1 = \sqrt{\frac{\rho\_1}{\lambda}} \quad \begin{array}{c} \\ \\ \hline \frac{D\_2}{\lambda} \end{array} \quad \begin{array}{c} \\ \\ \\ \hline \end{array} \tag{122}$$

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:

$$A\_1 e^{\delta\_{f\_1}} + B\_2 e^{-\delta\_{f\_1}} = \mathcal{C}\_o \qquad \text{at} \quad \mathbf{x} = -\delta$$

$$A\_1 + B\_1 = \mathcal{C}\_i \qquad \text{at} \quad \mathbf{x} = \mathbf{0}$$

The solutions of these equations are:

*Recent Techniques and Applications in Ionizing Radiation Research*

$$\begin{aligned} A\_1 &= \frac{e^{\delta/l\_1}}{e^{\delta/l\_1 - e^{-\delta/l\_1}}} C\_l - \frac{1}{e^{\delta/l\_1 - e^{-\delta/l\_1}}} C\_o \\ B\_1 &= \frac{e^{-\delta/l\_1}}{e^{\delta/l\_1 - e^{-\delta/l\_1}}} C\_l + \frac{1}{e^{\delta/l\_1 - e^{-\delta/l\_1}}} C\_o \end{aligned} \tag{124}$$

or

and

IF *δ*

*l*1 is small sinh *δ*

**7. The calibration factor**

measured by *Track cm*�2

**93**

1 day = 86,400 sec and 1 m3 = 106 cm3

*Kexperement*

*dC*2ð Þ *x dx*

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

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

> *l*1 � � ¼ *δ*

From define transmission factor we get as:

� � � � *x*¼0

*Ci* ¼ *Co* cosh *δ l*1 � � þ *D*2 *D*1 *l*1 *l*2 sinh *δ l*1 � � tanh *L*

*l*1 and cosh *δ*

*C* ¼

*Transmission* ð Þ¼ %

relation between *Ktheoretical* and *Kexperement* in the can technique is:

*Bq* � *m*�3 � *day* � � ¼ 0*:*0864 X

*Track cm*�2

*λ l*2*δ*

¼ � 1 *l*2

*l*1

*l*2 *L*

*D*1 þ coth *L*

*C Co* ¼

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, 17–19]. The theoretical calibration factor is calculated by unit (cm), while the experiment calibration factor is

*Bq*�*m*�3�*day* � �. Since 1 Bq = Disintegration/second = track/second,

, so, 1 cm = 0.0864 *Track cm*�2

*Ktheoretical*

*i*

tanh *L l*2

� � ¼ 1, we get:

*l*2

*λ l*2*δ*

*l*2 *L*

*D*1 þ coth *L*

*l*2

� �*Ci* (134)

� � (135)

� � (137)

*Bq* �*m*�3 �*day* � � [20]. The

*i* ð Þ *cm* (138)

*l*2

� �*Co* (136)

ð133Þ

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

$$\mathbf{C}\_{1}(\mathbf{x}) = \frac{\sinh\left(\frac{\mathbf{x} - \boldsymbol{\delta}}{l\_{1}}\right)}{\sinh\left(\frac{\boldsymbol{\delta}}{l\_{1}}\right)} \mathbf{C}\_{i} + \frac{\sinh\left(\frac{\boldsymbol{\chi}}{l\_{1}}\right)}{\sinh\left(\frac{\boldsymbol{\delta}}{l\_{1}}\right)} \mathbf{C}\_{o} \tag{125}$$

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

$$\mathcal{C}\_2(\boldsymbol{x} = \mathbf{0}) = \mathcal{C}\_l \Bigg\acity\_{\mathcal{C}\_2(\boldsymbol{x})} \Bigg\bf \Bigg\bf \begin{pmatrix} \frac{d\mathcal{C}\_2(\boldsymbol{x})}{d\boldsymbol{x}} \end{pmatrix}\_{\mathcal{X} = \boldsymbol{L}} = \mathbf{0} \end{pmatrix} \tag{126}$$

We get two equations:

$$\begin{array}{ll} A\_2 + B\_2 = \left. C\_l \right|\_{l\_2 = \frac{B\_2}{l\_2}} & \text{at} \quad \left. \begin{array}{l} \text{at} = 0 \\ \left. \begin{array}{l} \text{at} \end{array} \right|\_{\frac{B\_2}{d}} = 0 \end{array} \right| \\ \left. \begin{array}{l} \text{at} \end{array} \right|\_{\frac{B\_2}{d}} = 0 \end{array} \quad \text{at} \quad \left. \begin{array}{l} \text{at} = 0 \\ \left. \begin{array}{l} \text{at} \end{array} \right|\_{\frac{B\_2}{d}} = 0 \end{array} \right) \end{array} \tag{127}$$

The solutions of these equations are:

$$\begin{aligned} A\_2 &= \left. \frac{e^{-L\_{f\_{l\_2}}}}{e^{L\_{f\_{l\_2}}} + e^{-L\_{f\_{l\_2}}}} C\_i \right| \\ B\_2 &= \left. \frac{e^{L\_{f\_{l\_2}}}}{e^{L\_{f\_{l\_2}}} + e^{-L\_{f\_{l\_2}}}} C\_i \right] \end{aligned} \tag{128}$$

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

$$\mathbf{C}\_{2}(\mathbf{x}) = \frac{\sinh\left(\frac{\mathbf{x} - L}{l\_{2}}\right)}{\sinh\left(\frac{L}{l\_{2}}\right)} \mathbf{C}\_{i} \tag{129}$$

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

$$\overline{C} = \frac{1}{L} \int\_0^L C\_2(\varkappa) d\varkappa \tag{130}$$

The average concentration radon inter cup:

$$\overline{\mathbf{C}} = \frac{l\_2}{L} \tanh\left(\frac{L}{l\_2}\right) \mathbf{C}\_i \tag{131}$$

3.The boundary condition number 3 is:

$$\left.E\_1\right|\_{\mathbf{x}=\mathbf{0}} = \left.E\_2\right|\_{\mathbf{x}=\mathbf{0}}\tag{132}$$

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

or

ð124Þ

ð126Þ

ð127Þ

ð128Þ

(130)

� � *Ci* (129)

*Ci* (131)

*x*¼0 (132)

� �*Co* (125)

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

sinh *x*�*δ l*1 � � sinh *x l*1 � �

sinh *δ l*1

sinh *δ l*1 � � *Ci* þ

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

*C*1ð Þ¼ *x*

*Recent Techniques and Applications in Ionizing Radiation Research*

We get two equations:

The solutions of these equations are:

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

*C*2ð Þ¼ *x*

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

*C* ¼ 1 *L* ð*L* 0

*C* ¼ *l*2

*E*1j

The average concentration radon inter cup:

3.The boundary condition number 3 is:

**92**

sinh *x*�*L l*2 � �

sinh *L l*2

*C*2ð Þ *x dx*

*l*2 � �

*L* tanh *L*

*x*¼0 ¼ *E*2j

$$\begin{aligned} \left. -D\_1 \frac{d\mathcal{C}\_1(\boldsymbol{x})}{d\boldsymbol{x}} \right|\_{\boldsymbol{x}=0} &= \left. -D\_2 \frac{d\mathcal{C}\_2(\boldsymbol{x})}{d\boldsymbol{x}} \right|\_{\boldsymbol{x}=0} \\\\ \left. \frac{d\mathcal{C}\_1(\boldsymbol{x})}{d\boldsymbol{x}} \right|\_{\boldsymbol{x}=0} &= \frac{1}{l\_1} \left\{ \frac{\cosh\left(\frac{\delta}{l\_1}\right)}{\sinh\left(\frac{\delta}{l\_1}\right)} \mathcal{C}\_t - \frac{1}{\sinh\left(\frac{\delta}{l\_1}\right)} \mathcal{C}\_o \right\} \end{aligned} \tag{133}$$

and

$$\left. \frac{d\mathbf{C}\_2(\mathbf{x})}{d\mathbf{x}} \right|\_{\mathbf{x}=\mathbf{0}} = -\frac{1}{l\_2} \tanh\left(\frac{L}{l\_2}\right) \mathbf{C}\_i \tag{134}$$

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

$$C\_i = \frac{C\_o}{\cosh\left(\frac{\delta}{l\_1}\right) + \frac{D\_2}{D\_1}\frac{l\_1}{l\_2}\sinh\left(\frac{\delta}{l\_1}\right)\tanh\left(\frac{L}{l\_2}\right)}\tag{135}$$

IF *δ l*1 is small sinh *δ l*1 � � ¼ *δ l*1 and cosh *δ l*1 � � ¼ 1, we get:

$$\overline{C} = \frac{\frac{l\_2}{L}}{\frac{\lambda \cdot l\_2 \delta}{D\_1} + \coth\left(\frac{L}{l\_2}\right)} C\_o \tag{136}$$

From define transmission factor we get as:

$$Transmission\ (\text{\(\(6\)} = \frac{\overline{C}}{C\_o} = \frac{\frac{L}{L}}{\frac{\lambda \cdot l\_2 \delta}{D\_1} + \coth\left(\frac{L}{l\_2}\right)}\tag{137}$$

#### **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, 17–19]. 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 *Ktheoretical* and *Kexperement* in the can technique is:

$$K\_{\text{experiment}} \left(\frac{\text{Tract } cm^{-2}}{Bq \cdot m^{-3} \cdot day}\right) = 0.0864 \sum\_{i} K\_{i}^{theoretical} (cm) \tag{138}$$

### **8. Conclusion remarks**

**Author details**

**95**

Ali Farhan Nader Alrekabi

Iraq Oil Ministry/Basra Oil Company, Basra University, Basra, Iraq

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to<EMAIL_ADDRESS>
provided the original work is properly cited.

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

### **Acknowledgements**

I would like to express my deep appreciation to:

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

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

**8. Conclusion remarks**

concentration.

fraction.

factor.

**94**

**Acknowledgements**

• Department of physics

out of the container Eq. (10).

concentration as shown in **Table 1**.

are liner equations **Table 2**.

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

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

3.The radon concentration reaches the equilibrium state at the same time

4.There are three different quantities measured by track density, integrated radon concentration, average radon concentration and saturate radon

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

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

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

interest in ventilation factor when houses and buildings design.

I would like to express my deep appreciation to:

• My supervisor Prof. Dr. Abdul Ridha Hussein Subber

• Education College for Pure Science, Basra University

• Dr. Basim Almayahi and IntechOpen publishing

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

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

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

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

(t ffi 7T\_(1/2)) regardless of the volume of the container.

*Recent Techniques and Applications in Ionizing Radiation Research*

#### **Author details**

Ali Farhan Nader Alrekabi Iraq Oil Ministry/Basra Oil Company, Basra University, Basra, Iraq

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Nader AF. Theoretical and Experimental Study to Evaluate Radioactivity Applied on a Selected Area in Basra Governorate [thesis]. Basra, Iraq: University of Basra, Iraq; 2015

[2] Abd-Elzaher M. An overview on studying 222Rn exhalation rates using passive technique solid-state nuclear track detectors. American Journal of Applied Sciences. 2012;**9**(10):1653-1659

[3] Tufail M, Mirza SM, Mahmood A, Qureshi AA. Application of a closed-can technique for measuring radon exhalation from mine samples of Punjab, Pakistan. Journal of Environmental Radioactivity. 2000;**50**: 267-275

[4] Jiang H, Liangquan GE, Lin Y, Yi GU. Preliminary study on a regional radon concentration in surface soil prediction method. Progress in Nuclear Science and Technology. 2011;**1**:364-367

[5] Byun SH. Radioactivity: Radioisotopes and Radiation Methodology Lecture Notes. Canada: University of McMaster; 2014

[6] Saad AF, Abdalla YK, Hussein NA, Elyaseery IS. Radon exhalation rate from building material used on the Garyounis. University campus, Benghazi Libya,Turkish. Journal of Environmental Science and Engineering. 2010;**34**:67-74

[7] Eman SA, Nageeb SH, El-Sersy AR. U and Th determination in natural samples using CR-39 and LR-115 track detectors. World Journal of Nuclear Science and Technology. 2012;**2**:36-40

[8] Koo VSY, Yip CWY, Ho JPY, Nikezik D, Yu KN. Sensitivity of LR- 115 detector in diffusion chamber to Rn222 in the presence of Rn220. Applied Radiation and Isotopes. 2002;**56**:953-956 [9] Yu KN, Nikezic D. Long-Term measurements of radon progeny concentration with solid-state nuclear track detectors. Radiation Measurements. 2005;**40**:560-568

in cup dosimeter. Radiation Measurements. 2008;**43**:418-421

NHN. Numerical and analytical

factor for CR-39 detectors in the chamber diffusion by using Monte-Carlo method and the mean critical angle. Scholars Research Library. 2014;

**5**:23-30

2014;**3**:48-52

[17] Nader AF, Subber ARH, Al-Hashimi

*Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

calculations of efficiency and calibration

[18] Nader AF, Subber ARH, Al-Hashimi NHN. Mathematical calculation to find the best chamber and detector radii used for measurements the range of αparticle. The International Journal of Engineering & Science (THE IJES).

[19] Subber ARH, Al-Hashmi NHN, Nader AF, Jebur JH, Khodier MK. Construct as a simple radon chamber for

measurement of radon detectors calibration factors. Pelagia Research

[20] Ismail AH, Jaafar MS. Experimental measurement on CR-39 response for radon gas and estimation the optimum dimension of dosimeters for detection of radon. In: Proceeding of the 3ed Asian Physics Symposinum. Bandung,

Library. 2015;**6**:128-131

Indonesia; 2009

**97**

[10] Somogyi G, Paripas B, Varga Z. Measurement of radon, radon daughters and thoron concentration by multidetector devices. Nuclear Tracks and Radiation Measurements. 1984;**8**(1–4): 423-427

[11] Nader AF. The determination of equilibrium factor of radon and thoron using LR-115 type II detector in a selected area from Basra governorate, Iraq. Journal of Physics Conference Series. 2019;**1258**:012032

[12] UNSCEAR. United Nations scientific committee on the effects of atomic radiation, annex D, distribution. Applied Radiation and Isotopes. 1982; **54**:467-473

[13] Abo-Elmaged M, Mansy M, Eissa HM, El-Fiki MA. Major parameters affecting the calculation of equilibrium factor using SSNTD-measured track densities. Radiation Measurements. 2006;**41**:235-240

[14] Hosoda M, Tokonami S, Sorimachi A, Janik M, Ishikaw T, Yatabe Y, et al. Experimental system to evaluate the effective diffusion coefficient of radon. Review of Scientific Instruments. 2009;**80**:013501

[15] Nader AF, Juber JH, Subber ARH, Al-Hashimi NHN. The diffusion coefficient and calibration factors for indoor radon measurement in bare and twin cup modes. Journal of Zankoy Sulaimani-A. 2016;**18**(1):49-56

[16] Eappen KP, Sahoo BK, Ramachandran TV, Mayya YS. Calibration factor for thoron estimation *Mathematical Expressions of Radon Measurements DOI: http://dx.doi.org/10.5772/intechopen.92647*

in cup dosimeter. Radiation Measurements. 2008;**43**:418-421

**References**

2015

267-275

[1] Nader AF. Theoretical and Experimental Study to Evaluate Radioactivity Applied on a Selected Area in Basra Governorate [thesis]. Basra, Iraq: University of Basra, Iraq;

*Recent Techniques and Applications in Ionizing Radiation Research*

[9] Yu KN, Nikezic D. Long-Term measurements of radon progeny concentration with solid-state nuclear

[10] Somogyi G, Paripas B, Varga Z. Measurement of radon, radon daughters and thoron concentration by multidetector devices. Nuclear Tracks and Radiation Measurements. 1984;**8**(1–4):

[11] Nader AF. The determination of equilibrium factor of radon and thoron using LR-115 type II detector in a selected area from Basra governorate, Iraq. Journal of Physics Conference

Series. 2019;**1258**:012032

[12] UNSCEAR. United Nations scientific committee on the effects of atomic radiation, annex D, distribution. Applied Radiation and Isotopes. 1982;

[13] Abo-Elmaged M, Mansy M,

[14] Hosoda M, Tokonami S, Sorimachi A, Janik M, Ishikaw T, Yatabe Y, et al. Experimental system to

evaluate the effective diffusion coefficient of radon. Review of

[16] Eappen KP, Sahoo BK, Ramachandran TV, Mayya YS.

Scientific Instruments. 2009;**80**:013501

[15] Nader AF, Juber JH, Subber ARH, Al-Hashimi NHN. The diffusion coefficient and calibration factors for indoor radon measurement in bare and twin cup modes. Journal of Zankoy Sulaimani-A. 2016;**18**(1):49-56

Calibration factor for thoron estimation

Eissa HM, El-Fiki MA. Major parameters affecting the calculation of equilibrium factor using SSNTD-measured track densities. Radiation Measurements.

track detectors. Radiation Measurements. 2005;**40**:560-568

423-427

**54**:467-473

2006;**41**:235-240

[2] Abd-Elzaher M. An overview on studying 222Rn exhalation rates using passive technique solid-state nuclear track detectors. American Journal of Applied Sciences. 2012;**9**(10):1653-1659

[3] Tufail M, Mirza SM, Mahmood A, Qureshi AA. Application of a closed-can

Environmental Radioactivity. 2000;**50**:

[4] Jiang H, Liangquan GE, Lin Y, Yi GU. Preliminary study on a regional radon concentration in surface soil prediction method. Progress in Nuclear Science and Technology. 2011;**1**:364-367

Methodology Lecture Notes. Canada: University of McMaster; 2014

[6] Saad AF, Abdalla YK, Hussein NA, Elyaseery IS. Radon exhalation rate from building material used on the Garyounis. University campus, Benghazi Libya,Turkish. Journal of

[7] Eman SA, Nageeb SH, El-Sersy AR. U

Nikezik D, Yu KN. Sensitivity of LR- 115 detector in diffusion chamber to Rn222 in the presence of Rn220. Applied

Radiation and Isotopes. 2002;**56**:953-956

and Th determination in natural samples using CR-39 and LR-115 track detectors. World Journal of Nuclear Science and Technology. 2012;**2**:36-40

[8] Koo VSY, Yip CWY, Ho JPY,

**96**

[5] Byun SH. Radioactivity: Radioisotopes and Radiation

Environmental Science and Engineering. 2010;**34**:67-74

technique for measuring radon exhalation from mine samples of Punjab, Pakistan. Journal of

[17] Nader AF, Subber ARH, Al-Hashimi NHN. Numerical and analytical calculations of efficiency and calibration factor for CR-39 detectors in the chamber diffusion by using Monte-Carlo method and the mean critical angle. Scholars Research Library. 2014; **5**:23-30

[18] Nader AF, Subber ARH, Al-Hashimi NHN. Mathematical calculation to find the best chamber and detector radii used for measurements the range of αparticle. The International Journal of Engineering & Science (THE IJES). 2014;**3**:48-52

[19] Subber ARH, Al-Hashmi NHN, Nader AF, Jebur JH, Khodier MK. Construct as a simple radon chamber for measurement of radon detectors calibration factors. Pelagia Research Library. 2015;**6**:128-131

[20] Ismail AH, Jaafar MS. Experimental measurement on CR-39 response for radon gas and estimation the optimum dimension of dosimeters for detection of radon. In: Proceeding of the 3ed Asian Physics Symposinum. Bandung, Indonesia; 2009

**99**

Section 3

Applications in Ionizing

Radiation

## Section 3

## Applications in Ionizing Radiation

**101**

cancer [6–8].

(

**Chapter 7**

**Abstract**

presented.

palladium, rhodium

**1. Introduction**

is radioisotope production [2].

Production of the 103Pd via

Brachytherapy Seed

*Pooneh Saidi and Mahdi Sadeghi*

Cyclotron and Preparation of the

This study will briefly explain the production of 103Pd via cyclotron for brachy-

**Keywords:** brachytherapy, cyclotron, production, cross-section, excitation function,

Cyclotrons are charged particle circular accelerators. They are a type of particle accelerator that has many applications in nuclear physics, industry, technology, and medicine. They play an important role in medicine; for example, they are used for radiation therapy, production of medical radioisotopes, and biomedical research [1]. As a particle accelerator, one of the important uses of the cyclotron in medicine

For a long period, radioisotope production is basically done in nuclear reactors, but their availability is slowly decreasing, and due to some advantages of radioisotope production with the cyclotron, the development of particle accelerators started in the past century, so accelerator-based production facilities are growing, and

In this chapter, the production method for the radioisotope, Palladium-103

103Pd), via cyclotron is discussed. Palladium-103 with energy emission about 20 keV results in the rapid dose falloff with the distance which is suitable for low-dose-rate (LDR) brachytherapy [3]. For nearly 25 years, brachytherapy sources containing 103Pd have been clinically introduced and are in use [4, 5]. Sources containing 103Pd are most commonly used in the treatment of prostate and eye

various radioisotopes suitable for medical applications are produced.

therapy use. The excitation functions of 103Rh(p,n)103Pd and 103Rh(d,2n)103Pd reactions were calculated using ALICE/91, ALICE/ASH, and TALYS-1.2 codes and compared with published data. Production of 103Pd was done via 103Rh(p,n)103Pd nuclear reaction. The target was bombarded with 18 MeV protons at 200 μA beam current for 15 h. After irradiation and radiochemical separation of the electroplated rhodium target, at the optimum condition, 103Pd was absorbed into Amberlite®IR-93 resin. The preparation of the brachytherapy seed, which is loaded by the resin beads, has also been presented. At least, the method to determine the dosimetric parameters for the seed by experimental measurement has been

#### **Chapter 7**

## Production of the 103Pd via Cyclotron and Preparation of the Brachytherapy Seed

*Pooneh Saidi and Mahdi Sadeghi*

#### **Abstract**

This study will briefly explain the production of 103Pd via cyclotron for brachytherapy use. The excitation functions of 103Rh(p,n)103Pd and 103Rh(d,2n)103Pd reactions were calculated using ALICE/91, ALICE/ASH, and TALYS-1.2 codes and compared with published data. Production of 103Pd was done via 103Rh(p,n)103Pd nuclear reaction. The target was bombarded with 18 MeV protons at 200 μA beam current for 15 h. After irradiation and radiochemical separation of the electroplated rhodium target, at the optimum condition, 103Pd was absorbed into Amberlite®IR-93 resin. The preparation of the brachytherapy seed, which is loaded by the resin beads, has also been presented. At least, the method to determine the dosimetric parameters for the seed by experimental measurement has been presented.

**Keywords:** brachytherapy, cyclotron, production, cross-section, excitation function, palladium, rhodium

#### **1. Introduction**

Cyclotrons are charged particle circular accelerators. They are a type of particle accelerator that has many applications in nuclear physics, industry, technology, and medicine. They play an important role in medicine; for example, they are used for radiation therapy, production of medical radioisotopes, and biomedical research [1]. As a particle accelerator, one of the important uses of the cyclotron in medicine is radioisotope production [2].

For a long period, radioisotope production is basically done in nuclear reactors, but their availability is slowly decreasing, and due to some advantages of radioisotope production with the cyclotron, the development of particle accelerators started in the past century, so accelerator-based production facilities are growing, and various radioisotopes suitable for medical applications are produced.

In this chapter, the production method for the radioisotope, Palladium-103 ( 103Pd), via cyclotron is discussed. Palladium-103 with energy emission about 20 keV results in the rapid dose falloff with the distance which is suitable for low-dose-rate (LDR) brachytherapy [3]. For nearly 25 years, brachytherapy sources containing 103Pd have been clinically introduced and are in use [4, 5]. Sources containing 103Pd are most commonly used in the treatment of prostate and eye cancer [6–8].

#### **Table 1.**

*Examples of radioisotopes commonly used in brachytherapy.*

Brachytherapy is a form of treatment where a sealed radioactive source placed on or in the tissue/tumor is to be irradiated. With this method, a high dose can be delivered to the tumor with a rapid decrease in dose in the surrounding normal tissue. Brachytherapy sources are usually encapsulated; the encapsulated sources are placed within the body cavities close to the tumor, in a lumen of organs, or implanted into the tumor or placed over the tissue to be radiated.

Depending on the source dose rate, brachytherapy sources are classified into three categories [9]:

**Table 1** shows examples of radioisotopes commonly used in brachytherapy for the three mentioned categories [10].

This study will briefly explain the production of the 103Pd via cyclotron and will shortly describe some detail of 103Pd production such as production process, targetry, radiochemical separation, seed fabrication, and seed dosimetry method.

#### **2. Materials and methods**

#### **2.1 Cyclotron production of 103Pd**

There are different methods for the production of 103Pd via cyclotron. Two of them are 103Rh(p,n)103Pd and 103Rh(d,2n)103Pd [3]. In this study, the excitation functions for both reactions have been calculated, and the optimum condition for each reaction has been obtained. Experimental data for the proton bombardment on rhodium metal as the target, via 103Rh(p,n)103Pd reaction, has also been measured. Rhodium target has been bombarded by proton in a cyclotron (Cyclone-30, IBA)

**103**

sufficient.

*Production of the 103Pd via Cyclotron and Preparation of the Brachytherapy Seed*

with 18 MeV energy and a beam current intensity of 200 μA at the Agricultural,

To achieve the 103Pd from the irradiated target, the radiochemical separation stage is started. The problem in this stage is the dissolution of target material due to the extremely low chemical reactivity of rhodium metal. The other problem is the high quantity of rhodium in the solution. A well-known palladium extractor is dimethylglyoxime, but to prevent the decrease of extraction yield, α-furyldioxime has been used [12]. Purely obtained 103Pd is then absorbed into resin; the active

Excitation functions of the 103Rh(p,n)103Pd and 103Rh(d,2n)103Pd reactions were calculated using ALICE/ASH, EMPIRE (version 3.1 Rivoli) and TALYS-1.2 nuclear codes, and the TENDL-2010. Using the codes simultaneously increases the accuracy of calculations. The calculated results were compared to the existing data of

The ALICE/ASH code: This code is a modified version of the ALICE code, and to describe the pre-equilibrium particle emission from nuclei, the geometry-dependent hybrid model (GDH) is used. Calculations were carried out based on the Fermi gas model with the nuclear level density parameter a = A/y and the generalized

The TALYS code: TALYS is a computer code developed at NRG Petten and CEA to predict and analyze the nuclear reactions. TALYS models the nuclear reactions that involve protons, deuterons, neutrons, alpha particles, gamma rays, tritons, and hellions. The code simulated the reactions in the energy range from 1 keV to

EMPIRE: EMPIRE (version 3.1 Rivoli) simulates various nuclear reactions, over a broad range of energies and incident particles. This system can be used for nuclear data evaluation and also for the theoretical calculation of nuclear reactions. A projectile can be a photon, a nucleon, and light or heavy ion. There is a broad range of energy in the system; the energy range starts just above the resonance region in the case of a neutron projectile and extends up to a few hundred MeV for heavy

The required thickness of the target has been calculated via the stopping and range of ions in matter (SRIM) code [21]. Based on the code results, to take full advantage of the excitation function and also to avoid the production of the radioisotope impurity, the entrance energy of the proton should be 18 MeV. The physical thickness of the rhodium layer is chosen in such a way that for a given beam/ target angle geometry, the particle exit energy should be 6 MeV. The thickness of the rhodium target has to be 475 μm for 90° geometry. To minimize the thickness of the rhodium layer (and hence lower the cost price per target), a 6° geometry is preferred; in this case, the thickness of the target decreases, and a 45–50 μm layer is

Identification of the gamma ray emitting from the radionuclides is performed by using gamma-ray spectroscopy with a high-purity germanium HP(Ge) detector

*DOI: http://dx.doi.org/10.5772/intechopen.92457*

Medical and Industrial Research School (AMIRS) [11].

resins are encapsulated inside the titanium casing.

**2.3 Nuclear models applied for cross-section calculations**

superfluid nuclear model. The default value of y is equal to [17, 19].

200 MeV, for target nuclides of mass 12 and heavier [14, 17].

**2.2 Calculation of excitation function**

references [13–18].

ion-induced reactions [20].

**2.4 The thickness of the target**

(Canberra™ model GC1020-7500SL).

*Production of the 103Pd via Cyclotron and Preparation of the Brachytherapy Seed DOI: http://dx.doi.org/10.5772/intechopen.92457*

with 18 MeV energy and a beam current intensity of 200 μA at the Agricultural, Medical and Industrial Research School (AMIRS) [11].

To achieve the 103Pd from the irradiated target, the radiochemical separation stage is started. The problem in this stage is the dissolution of target material due to the extremely low chemical reactivity of rhodium metal. The other problem is the high quantity of rhodium in the solution. A well-known palladium extractor is dimethylglyoxime, but to prevent the decrease of extraction yield, α-furyldioxime has been used [12]. Purely obtained 103Pd is then absorbed into resin; the active resins are encapsulated inside the titanium casing.

#### **2.2 Calculation of excitation function**

*Recent Techniques and Applications in Ionizing Radiation Research*

Brachytherapy is a form of treatment where a sealed radioactive source placed on or in the tissue/tumor is to be irradiated. With this method, a high dose can be delivered to the tumor with a rapid decrease in dose in the surrounding normal tissue. Brachytherapy sources are usually encapsulated; the encapsulated sources are placed within the body cavities close to the tumor, in a lumen of organs, or

**Radionuclide T1/2 Mode of decay The energy of the emitted particle (keV)**

32P 14.26 d β−(100) 690 137Cs 30.04 y β−(100) 662 198Ir 73.8 β−(95.34) 380 198Au 2.69 β−(100) 412 125I 59.4 d EC (100) 28 169Yb 32.02 d EC (100) 93 103Pd 16.99 d EC (100) 21 153Sm 46.28 h β−(100) 223 142Pr 19.12 β−(96.3) 809 170Tm 128.6 d β−(99.87) 66.39

Depending on the source dose rate, brachytherapy sources are classified into

• High-dose-rate sources (HDR): >12 Gy/h radioisotopes with high-energy

• Medium-dose-rate sources (MDR): 2–12 Gy/h radioisotopes are used; this

• Low-dose-rate sources (LDR): Less than 2 Gy/h radioisotopes with low-energy

**Table 1** shows examples of radioisotopes commonly used in brachytherapy for

This study will briefly explain the production of the 103Pd via cyclotron and will shortly describe some detail of 103Pd production such as production process, targetry, radiochemical separation, seed fabrication, and seed dosimetry method.

There are different methods for the production of 103Pd via cyclotron. Two of them are 103Rh(p,n)103Pd and 103Rh(d,2n)103Pd [3]. In this study, the excitation functions for both reactions have been calculated, and the optimum condition for each reaction has been obtained. Experimental data for the proton bombardment on rhodium metal as the target, via 103Rh(p,n)103Pd reaction, has also been measured. Rhodium target has been bombarded by proton in a cyclotron (Cyclone-30, IBA)

implanted into the tumor or placed over the tissue to be radiated.

*Examples of radioisotopes commonly used in brachytherapy.*

photon emitters like 137Cs, 60Co, 192Ir, and 198Au are used.

category is not commonly used in brachytherapy.

photon emitters like 103Pd and 125I are used.

the three mentioned categories [10].

**2. Materials and methods**

**2.1 Cyclotron production of 103Pd**

three categories [9]:

**Table 1.**

**102**

Excitation functions of the 103Rh(p,n)103Pd and 103Rh(d,2n)103Pd reactions were calculated using ALICE/ASH, EMPIRE (version 3.1 Rivoli) and TALYS-1.2 nuclear codes, and the TENDL-2010. Using the codes simultaneously increases the accuracy of calculations. The calculated results were compared to the existing data of references [13–18].

#### **2.3 Nuclear models applied for cross-section calculations**

The ALICE/ASH code: This code is a modified version of the ALICE code, and to describe the pre-equilibrium particle emission from nuclei, the geometry-dependent hybrid model (GDH) is used. Calculations were carried out based on the Fermi gas model with the nuclear level density parameter a = A/y and the generalized superfluid nuclear model. The default value of y is equal to [17, 19].

The TALYS code: TALYS is a computer code developed at NRG Petten and CEA to predict and analyze the nuclear reactions. TALYS models the nuclear reactions that involve protons, deuterons, neutrons, alpha particles, gamma rays, tritons, and hellions. The code simulated the reactions in the energy range from 1 keV to 200 MeV, for target nuclides of mass 12 and heavier [14, 17].

EMPIRE: EMPIRE (version 3.1 Rivoli) simulates various nuclear reactions, over a broad range of energies and incident particles. This system can be used for nuclear data evaluation and also for the theoretical calculation of nuclear reactions. A projectile can be a photon, a nucleon, and light or heavy ion. There is a broad range of energy in the system; the energy range starts just above the resonance region in the case of a neutron projectile and extends up to a few hundred MeV for heavy ion-induced reactions [20].

#### **2.4 The thickness of the target**

The required thickness of the target has been calculated via the stopping and range of ions in matter (SRIM) code [21]. Based on the code results, to take full advantage of the excitation function and also to avoid the production of the radioisotope impurity, the entrance energy of the proton should be 18 MeV. The physical thickness of the rhodium layer is chosen in such a way that for a given beam/ target angle geometry, the particle exit energy should be 6 MeV. The thickness of the rhodium target has to be 475 μm for 90° geometry. To minimize the thickness of the rhodium layer (and hence lower the cost price per target), a 6° geometry is preferred; in this case, the thickness of the target decreases, and a 45–50 μm layer is sufficient.

Identification of the gamma ray emitting from the radionuclides is performed by using gamma-ray spectroscopy with a high-purity germanium HP(Ge) detector (Canberra™ model GC1020-7500SL).

### **2.5 Preparation of 103Pd brachytherapy seeds**

After irradiation, the radiochemical separation phase has been started. In this phase, the PdCl2 solution has been separated from rhodium, zinc, and copper.

According to the brachytherapy seed model (in case of using a sphere made of resin), resin beads and marker are encapsulated inside the titanium capsule. The end caps of the capsule are welded precisely to prevent source leakage [22]. Regarding the physical design and configuration of the source internal component, two types of designs have been used: (a) rod/wire/cylinder made of ceramic, glass, or high-Z materials and (b) sphere made of resin. In this study sphere design of the source has been discussed.

**Figure 1** shows a schematic diagram of eight different brachytherapy seeds which are designed at the Agricultural, Medical, and Industrial Research School [7, 23–25].

#### **2.6 Dosimetry of the seed**

According to the American Association of Physicists in Medicine (AAPM) Radiation Therapy Committee recommendation, the dosimetry characteristics

**105**

**Figure 2.**

*experimental data [10].*

*Production of the 103Pd via Cyclotron and Preparation of the Brachytherapy Seed*

for all new interstitial brachytherapy seeds with energies less than 50 keV should be investigated by two independent investigators, theoretical calculations and experimental measurements. This work presents the method for thermoluminescent dosimeter (TLD) measurements to determine the dosimetric characteristics of the brachytherapy seed containing resin beads. The TLD-GR200A thermoluminescent dosimeters and two Perspex phantoms have been used, one Perspex phantom for the anisotropy function, *F(r, θ)*, and the other for the radial dose

The evaluation of the acquired data from the codes showed that the best range of the energy for proton in the 103Rh(p,n)103Pd reaction is 18–8 MeV. The maximum cross-section by EMPIRE (version 3.1 Rivoli) code is at Ep = 10 MeV, and the value is 574.44 mb. To evaluate the obtained results, **Figure 2** shows the comparison between the calculated results in this study and measured data by others. For the 103Rh(p,n)103Pd reaction, there are five cross-section measurements that exist in the

The calculated results from the TENDL-201, TALYS-1.2, and ALICE/ASH codes

According to the results from the codes, the optimum range of energy of the deuteron particle to produce 103Pd from 103Rh target for the 103Rh(d,2n)103Pd reaction is 22 to 8 MeV. The obtained results from ALICE/ASH hybrid model (a = A/9)

*Excitation function of 103Rh(p,n)103Pd reaction by ALICE/91, ALICE/ASH and TALYS-1.2 codes, and*

are in acceptable agreement with the measured data from [29], and calculated results from EMPIRE (version 3.1 Rivoli) code are in good agreement with the

*DOI: http://dx.doi.org/10.5772/intechopen.92457*

function, *gL(r)*.

**3. Results and discussion**

*3.1.1 Excitation function study of 103Rh(p,n)103Pd reaction*

literature by authors of references [26–29].

Hermanne et al. measured data (**Figure 2**).

*3.1.2 Excitation function study of 103Rh(d,2n)103Pd reaction*

**3.1 Excitation function**

**Figure 1.**

*Schematic drawing of the designed 103Pd sources.*

*Production of the 103Pd via Cyclotron and Preparation of the Brachytherapy Seed DOI: http://dx.doi.org/10.5772/intechopen.92457*

for all new interstitial brachytherapy seeds with energies less than 50 keV should be investigated by two independent investigators, theoretical calculations and experimental measurements. This work presents the method for thermoluminescent dosimeter (TLD) measurements to determine the dosimetric characteristics of the brachytherapy seed containing resin beads. The TLD-GR200A thermoluminescent dosimeters and two Perspex phantoms have been used, one Perspex phantom for the anisotropy function, *F(r, θ)*, and the other for the radial dose function, *gL(r)*.

#### **3. Results and discussion**

#### **3.1 Excitation function**

*Recent Techniques and Applications in Ionizing Radiation Research*

After irradiation, the radiochemical separation phase has been started. In this phase, the PdCl2 solution has been separated from rhodium, zinc, and copper. According to the brachytherapy seed model (in case of using a sphere made of resin), resin beads and marker are encapsulated inside the titanium capsule. The end caps of the capsule are welded precisely to prevent source leakage [22]. Regarding the physical design and configuration of the source internal component, two types of designs have been used: (a) rod/wire/cylinder made of ceramic, glass, or high-Z materials and (b) sphere made of resin. In this study sphere design of the

**Figure 1** shows a schematic diagram of eight different brachytherapy seeds which are designed at the Agricultural, Medical, and Industrial Research School

According to the American Association of Physicists in Medicine (AAPM) Radiation Therapy Committee recommendation, the dosimetry characteristics

**2.5 Preparation of 103Pd brachytherapy seeds**

source has been discussed.

**2.6 Dosimetry of the seed**

[7, 23–25].

**104**

**Figure 1.**

*Schematic drawing of the designed 103Pd sources.*

#### *3.1.1 Excitation function study of 103Rh(p,n)103Pd reaction*

The evaluation of the acquired data from the codes showed that the best range of the energy for proton in the 103Rh(p,n)103Pd reaction is 18–8 MeV. The maximum cross-section by EMPIRE (version 3.1 Rivoli) code is at Ep = 10 MeV, and the value is 574.44 mb. To evaluate the obtained results, **Figure 2** shows the comparison between the calculated results in this study and measured data by others. For the 103Rh(p,n)103Pd reaction, there are five cross-section measurements that exist in the literature by authors of references [26–29].

The calculated results from the TENDL-201, TALYS-1.2, and ALICE/ASH codes are in acceptable agreement with the measured data from [29], and calculated results from EMPIRE (version 3.1 Rivoli) code are in good agreement with the Hermanne et al. measured data (**Figure 2**).

#### *3.1.2 Excitation function study of 103Rh(d,2n)103Pd reaction*

According to the results from the codes, the optimum range of energy of the deuteron particle to produce 103Pd from 103Rh target for the 103Rh(d,2n)103Pd reaction is 22 to 8 MeV. The obtained results from ALICE/ASH hybrid model (a = A/9)

#### **Figure 2.**

*Excitation function of 103Rh(p,n)103Pd reaction by ALICE/91, ALICE/ASH and TALYS-1.2 codes, and experimental data [10].*

show that the maximum cross-section is 1158.795 mb (Ed = 13 MeV). Comparison between the calculated results of this study and the measured data obtained by [30, 31] is presented in **Figure 3**.

The obtained results from ALICE/ASH code are in good agreement with the measured data by Hermanne et al. up to 20 MeV, whereas TALYS-1.2 and EMPIRE (version 3.1 Rivoli) calculated results have lower values than ALICE/ASH results and also experimental data.

#### **3.2 Production of 103Pd**

To prepare the rhodium target for irradiation, via the electrodeposition process, a thick layer of rhodium has been placed of the copper backing. According to Sadeghi et al. study [11, 12], the following conditions are the optimum conditions for the electrodeposition:

After the electrodeposition process, the adhesion quality of the rhodium layer on the target backing has been tested by the thermal shock. The thermal shock has been carried out by heating the target up to 500°C for 1 h (The temperature that the Rh layer experience during high current irradiation). Thereafter, the hot target is submerged in cold water in a temperature of about 15°C. Observation of neither crack formation nor peeling off of the rhodium layers indicated a good adhesion for the purpose.

Afterward, the rhodium target was bombarded with 18 MeV protons at 200 μA beam current for 15 h (3000 μAh) [12]. At the end of the bombardment (EOB),

#### **Figure 3.**

*Excitation function of 103Rh(d,2n)103Pd reaction by ALICE/91, ALICE/ASH and TALYS-1.2 codes, and experimental data [10].*

**107**

**Figure 4.**

*Production of the 103Pd via Cyclotron and Preparation of the Brachytherapy Seed*

The optimum conditions of the electro-dissolution are as follows:

the activity of 103Pd and the yield of production are 685 mCi and 8.44 MBq/lAh, respectively. After the proton bombardment, the dissolution process has been started.

After the dissolution of the target from copper backing, the residual contains PdCl2, rhodium, zinc, and copper, so during the radiochemical separation phase, the PdCl2 solution should be separated from rhodium, zinc, and copper. According to the data in **Figure 4**, the purity of the obtained radio-palladium is about 99%. Thereafter, the obtained filtrate solution was loaded onto the column of size Ø 0.5 cm × 2 cm packed with Amberlite®IR-93 resin with 0.6 mm diameter. The summarized results in **Figure 5** show that 0.05 M HCl is the most suitable concentration

The dosimetric parameters of the seeds have been determined by theoretical calculation and experimental measurement, according to TG-43 U1 report. The theoretical method to obtain the dosimetric parameters of the brachytherapy seeds

The following is the method to determine the dosimetric parameters by experi-

The TLD-GR200A (PTW, Freiburg, Germany) circular chips [26] of the follow-

*DOI: http://dx.doi.org/10.5772/intechopen.92457*

• AC current density > 1.8 A cm2

for adsorption of 103Pd on the Amberlite®IR-93 resin.

has been discussed and explained in Refs. [6, 9].

ing specifications have been used in this study:

*HPGe spectrum of radiochemically separated 103Pd [10].*