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A model to explain simultaneously the 222Rn and 220Rn emanation from thin electrodeposited sources
Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
A model to explain simultaneously the Rn and Rn emanation from thin electrodeposited sources M. Jurado Vargas* Departamento de Fn& sica, Universidad de Extremadura, 06071 Badajoz, Spain Received 13 July 1999; received in revised form 5 November 1999; accepted 29 November 1999
Abstract In thin radioactive sources, loss of radon by emanation is a very common phenomenon, especially in sources made by electrodeposition. A quanti"cation of this e!ect in radium sources can be easily developed by using a simple model that assumes a radon di!usion term in the ingrowth equations. By measuring the corresponding Rn/Ra activity ratio, a constant di!usion factor can be determined which represents the Rn emanation from the whole source. However, this simple model cannot explain simultaneously the Rn and Rn di!usion produced in a thin source, because it gives di!usion factors that are di!erent by many orders of magnitude for these two isotopes, while these values must be fairly close. In this paper, a new model of di!usion is proposed, which includes a linear dependence of the di!usion factor on the depth of Rn nuclides in the source. This new model has been applied to radium electrodeposited sources and allows us to explain satisfactorily both the Rn/Ra and Rn/Ra activity ratios observed in thin sources. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Several techniques have been described in the literature for the determination of radium in environmental samples. These include gamma ray spectrometry, liquid scintillation counting, radon emanation and alpha particle spectrometry using the coprecipitation of radium with barium sulphate [1,2]. Although this last technique has given satisfactory results for radium determination, since the losses of radon by di!usion from the source are then minimal mainly due to the thickness of the Ba(Ra)SO layer, it o!ers poor energy resolutions. This problem has been solved using procedures for
* Tel.: #34-24-289300; fax: #34-24-275428. E-mail address: [email protected] (M.J. Vargas).
radium electrodeposition [3}7], which allow one to achieve thinner layers and improved spectra with better energy resolutions, but also involve losses of radon from the sources. In previous work, we have performed some studies about Rn di!usion in Ra sources made by electrodeposition [8,9]. The loss of radon by emanation was observed by measuring the experimental Rn/Ra activity ratio versus the time elapsed from the Ra source preparation, and a quanti"cation of this phenomenon was developed by using a simple model that assumes a radon di!usion term in the ingrowth di!erential equations. In this way, a constant radon di!usion coef"cient D was deduced for each Ra source. These values ranged approximately between 0.10 and 0.23 days\ for a great number of Ra electrodeposited sources.
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 2 8 9 - 9
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
In spite of the good behaviour of this simple model in explaining the Rn di!usion, it however cannot explain simultaneously the Rn and Rn di!usion produced in a thin source, because it gives di!usion factors that are di!erent by many orders of magnitude for these two isotopes, while these values must be fairly close. In this paper, a more realistic model of di!usion is proposed, which includes a linear dependence of the di!usion factor on the depth of Rn nuclides in the source. This new model has been applied to radium electrodeposited sources and allows us to explain satisfactorily both Rn/Ra and Rn/Ra activity ratios observed in thin sources.
2. Constant di4usion model In order to describe the radon emanation e!ect, the simple model introduces a di!usion term into the ingrowth di!erential equations. This term represents the radon emanation rate and is proportional to the number of radon nuclides present at each time, with a constant di!usion factor D. Hence, the corresponding di!erential equation can be written as dN(Rn)"j(Ra)N(Ra) dt![j(Rn)#D]N(Rn)dt. (1) By solving this di!erential equation, one "nds that the activity ratio Rn/Ra at each time is given by
Rn j(Rn) (t)" Ra j(Rn)!j(Ra)#D) ;[1!exp([j(Rn)!j(Ra)#D]t)]. (2)
This expresion may be applied to both the Rn/Ra or Rn/Ra activity ratios. If we take into account the decay constants k of Rn, Ra, Rn and Ra (2.098;10\; 1.374;10\; 0.01247 and 2.192;10\ s\, respectively), the exponential term in Eq. (2) will be negligible for su$ciently large times after the radium source preparation, so that the Rn/Ra and Rn/Ra activity ratios will
609
reach a constant value given by
j(Rn) Rn " . Ra j(Rn)!j(Ra)#D
(3)
The Rn/Ra ratio will reach the constant value given by expression (3) at a time of about 20 days after the source preparation, while the Rn/Ra ratio will reach its constant value in a far shorter time of the order of a few minutes, due to the much smaller half-lives of Rn and Ra. In other words, the exponential term in the Rn/Ra ratio does not fall to zero immediately as e!ectively occurs for the Rn/Ra ratio. In previous work [8,9] we have observed a significant Rn emanation from Ra sources prepared by using some radium electrodeposition methods [5,6], as seen in the alpha-particle spectrum of electrodeposited Ra shown in Fig. 1 for example. Using the model of constant di!usion, the experimental Rn/Ra activity ratios were "tted versus time to the theoretical Eq. (2). Table 1 lists some typical values found for the Rn/Ra activity ratio, which range from 0.41 to 0.56, and the values of the constant di!usion factor D calculated for each source, which range typically between 0.142 and 0.261 days\ (see third column in Table 1). Also, several authors have reported signi"cant Rn retentions in thin radium sources and samples of soils and rocks [10}13], so that Rn di!usion does not allow the presumption of secular equilibrium of Rn with its parent Ra. In particular, Hanckock and Martin [13] observed typical Rn retentions that ranged from 0.57 to 0.77 for radium electrodeposited sources by using the same method as ours. Sill [10] noted that the Rn retention of a thin radium source prepared from a "ne precipitate of barium sulphate was 58%. Taking into consideration that the di!usion coef"cient D for Rn has to be fairly close to that of Rn, we can calculate the theoretical values of Rn retention that would be obtained in the source, by using as di!usion coe$cient in equation (3) with the value obtained for Rn. These values for Rn retentions are listed in the fourth column of Table 1. As seen in this table, these theoretical values con"rm the theoretical secular equilibrium
610
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
Fig. 1. Alpha-particle spectrum of Ra obtained from a standard solution by using the electrodeposition procedure proposed by Whitehead et al. [6]. Note the high degree of disequilibrium between Rn and Ra, indicating a major di!usion of Rn from the source. Table 1 Experimental Rn/Ra activity ratios observed by the author in some radium electrodeposited sources (after 30 days following the electrodeposition) and the values of the Rn constant di!usion coe$cient D calculated for each Ra source (third column). The fourth column gives the theoretical value of the Rn/Ra activity ratio that would be obtained in the source by assuming the calculated Rn di!usion coe$cient for Rn in Eq. (3). Numbers in parentheses denote the reference for the electrodeposition procedure Electrodeposition Observed procedure Rn/Ra
Calculated D [days\]
Calculated Rn/Ra
[6] [5] [6] [6] [5] [5] [6] [5],[6] [5]
0.261 0.250 0.240 0.213 0.204 0.196 0.167 0.148 0.142
0.999758 0.999768 0.999777 0.999802 0.999811 0.999818 0.999845 0.999863 0.999868
0.41 0.42 0.43 0.46 0.47 0.48 0.52 0.55 0.56
between Rn and Ra. The reason for this result is that, although Rn (half-life"3.8 days) is di!used by emanation from the source, the di!usion of Rn must be practically null due to its short half-life (3.96 s). In other words, for this last nuclide, the decay rate would be much higher than any possible emanation rate, so that there will be negligible Rn emerging from the source. Obviously, these results are not in agreement with the experimental values found in the literature for radium electrodeposited sources, which involve signi"cant Rn retentions. We have already reported this disagreement in the constant di!usion model in a previous paper [14].
3. Linear di4usion model A new model is proposed. Firstly, it is assumed that Ra and Rn nuclides in the source are uniformly
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
611
Although the mean Rn di!usion length is high in soils and other materials, there are evidences for a practically null di!usion length in some thin sources, where the Rn losses are minimal [9]. These results are in accordance with the existence of a null di!usion zone, as established here. The Rn di!usion coe$cient will then have the following values:
D(x)" Fig. 2. Simple distribution model of Ra and Rn nuclides in the source. Nuclei are deposited uniformly to a thickness L in the source.
distributed, so that nuclides are `incrusteda homogeneously to a depth `¸a into the source, as shown schematically in Fig. 2. In this way, a constant distribution function for Rn nuclides is obtained, which normalized is given by F"1/¸. Secondly, the constant Rn di!usion coe$cient introduced in the simple model must be substituted by a new, more realistic coe$cient that exhibts a dependence on the depth of Rn nuclides in the source. The situation is illustrated in Fig. 3, where Rn atoms that are deeper in the source (zone from 0 to a¸) are assumed not to escape via di!ussion, while the rest undergo linear di!usion potential reaching a maximum value D for the atoms located at the surface of the source.
Fig. 3. Schematic illustration of the linear di!usion model. Rn nuclides that are deeper in the source do not di!use, while the rest undergo linear di!usion relative to the location of the Rn nuclides in the source.
0
aD D x! ¸(1!a) (1!a)
for 0(x4a¸, for a¸4x4¸.
(4)
The Rn/Ra activity ratio must then be given by the contributions of all the nuclei located at di!erent depths in the source, which have di!erent values of the di!usion coe$cient. Taking into consideration (3) for a constant di!usion coe$cient D(x), the Rn/Ra activity ratio in this linear model will be given by
* [Rn/Ra(x)]F(x) dx * j(Rn) " dx. (5) [j(Rn!j(Ra)#D(x)]¸ If we take into account that the radioactive constants j(Ra) and j(Ra) are several orders of magnitude smaller than j(Rn) and j(Rn), respectively, integration of Eq. (6) will give (Rn/Ra)"
(1!a) j(Rn) D (Rn/Ra)"a# In 1# . (6) j(Rn) D It must be noted that Eq. (6) can be applied to both Rn/Ra and Rn/Ra activity ratios. The proposed linear di!usion model gives a Rn/Ra activity ratio that depends only on two parameters, a and D . The "rst parameter, a corres ponds to the percentage of Rn nuclides in the source that do not undergo any di!usion due to their location at the deepest layers of deposition. The second parameter, D , corresponds to the di!u sion coe$cient for Rn nuclides located on the surface of the source, representing the maximum value of Rn di!usion. These two parameters are only related to the di!usion phenomenon and therefore have to be common to both Rn and Rn isotopes in a given source.
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M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
Table 2 Values obtained for the a parameter and values deduced for the Rn/Ra activity ratio by using the proposed linear di!usion model. Three typical values for the experimental Rn/Ra activity ratio obtained in radium electrodeposited sources and several values of the surface di!usion coe$cient D have been taken Assumed values
Calculated values
D (days\)
Rn/Ra
a
Rn/Ra
1
0.40 0.48 0.56
0.0896 0.2110 0.3324
0.999578 0.999634 0.999690
10
0.40 0.48 0.56
0.3525 0.4388 0.5252
0.997013 0.997411 0.997810
50
0.40 0.48 0.56
0.3875 0.4691 0.5508
0.986210 0.988047 0.989887
100
0.40 0.48 0.56
0.3930 0.4739 0.5549
0.973456 0.976994 0.980536
1000
0.40 0.48 0.56
0.3991 0.4792 0.5593
0.824146 0.847588 0.871029
5000
0.40 0.48 0.56
0.3998 0.4798 0.5598
0.623524 0.673704 0.723884
10 000
0.40 0.48 0.56
0.3999 0.4799 0.5599
0.500293 0.610468 0.670385
This linear model then has to explain both Rn and Rn retentions in a thin source. In order to check the model, several values for the Rn/Ra activity ratio and the surface di!usion coe$cient D were assumed (see Table 2). Three typical values of Rn/Ra were assumed, in agreement with those reported above in this present work in Table 1. From these values for the Rn/Ra ratio and the D coe$cient, the values for the a parameter were obtained, and hence the theoretical Rn/Ra activity ratios deduced. As seen in Table 2, the values obtained for the parameter a become very close to the Rn/Ra
ratio as the value of D increases. This means that, when a high rate of di!usion is involved, the Rn/Ra ratio observed in thin sources is essentially due to the percentage of Rn nuclides that do not di!use in the source (which is quanti"ed in the parameter a), while the rest of the Rn nuclides practically do not contribute to Rn retention due to their great degree of di!usion from the source. In other words, for these more super"cial Rn nuclides the emanation rate is very much greater than the decay rate; i.e., D is very much greater than j(Rn). In addition, an observation of Table 2 shows that high Rn retentions can only be obtained by assuming high values for the surface di!usion coef"cient D . In particular, Rn/Ra ratios in the range observed by Hanckock and Martin [13] for thin electrodeposited sources (values that ranged from 0.569 to 0.771) can be obtained for the three assumed values of the Rn/Ra ratio by using surface coe$cients D greater than 1000 days\. In this case, then, the Rn/Ra activity ratio is due to the sum of two contributions: a "rst term given by the value of the parameter a, due to the Rn nuclides located in the deepest layers that do not undergo di!usion, while the second term is given by the more super"cial Rn nuclides, which also contribute to the Rn/Ra activity ratio, because the decay rate for Rn is signi"cant and cannot be hidden by the di!usion rate. In other words, here j(Rn) cannot be neglected versus D . In summary, as seen from Table 2, the linear di!usion model allows one to explain both Rn and Rn retentions in thin sources, by using the same function D(x) for the Rn di!usion coe$cient. The high Rn and Rn retentions observed in thin alpha sources can only be explained by assuming a steep slope in the D(x) function, i.e., very high values of the surface coe$cient D . In particular, for electrodeposited sources, this model establishes a parameter a given by the value of the Rn/Ra activity ratio and values of D of the order of 10}10 days\. Finally, an important point must be taken into consideration. The parameter a must be related to the fraction of Rn nuclides which di!use through the solid deposit and reach the sample pores. Therefore, it may be expected that the parameter
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
a for Rn and Rn would be di!erent, due to the di!erences in their half-lives. A new model based on the present one must be elaborated to account for this aspect. 4. Conclusions The loss of Rn by emanation in thin radioactive sources can be easily quanti"ed by assuming a Rn emanation term in the ingrowth equations, with a constant di!usion coe$cient. However, this simple model cannot explain both Rn and Rn emanations simultaneously, due to the very di!erent half-lives of these isotopes. In this present paper, a new model has been proposed which assumes two di!erent zones in the radioactive source: a "rst zone very deep in the source where Rn di!usion is null, and a second one, less deep, with a linear dependence of the di!usion factor on the depth of Rn nuclides in the deposit. In particular, the percentage of Rn nuclides in the "rst zone of the source is given by the observed Rn/Ra activity ratio, while high values of the surface coe$cient D (about 10}10 days\) must be involved in thin sources to derive the observed Rn retentions. This new model allows one to explain satisfactorily the high Rn and Rn retentions observed in thin sources, using the same function for the di!u-
613
sion coe$cient for both isotopes Rn and Rn. Thus, this model gives a more satisfactory explanation of the Rn emanation phenomenon observed in thin alpha sources. In spite of this, new models based on the present model must be proposed to allow di!erent values of the parameter a for Rn and Rn.
References [1] J. Sedlet, Radon and Radium. Treatise on Analytical Chemistry, Interscience, New York, 1966. [2] B. Keisch, A.S. Levine, Anal. Chem. 38 (1966) 1969. [3] D. Roman, Int. J. Appl. Radiat. Isotopes 35 (1984) 990. [4] S.A. Short, Nucl. Instr. Meth. B 17 (1986) 540. [5] G.J. Hancock, P. Martin, Appl. Radiat. Isot. 42 (1991) 63. [6] N.E. Whitehead, R.G. Ditchburn, W.J. McCabe, R. Van der Raaij, J. Radioanal. Nucl. Chem. Art. 160 (1992) 477. [7] T.F. Hamilton, V.M. McRae, J.D. Smith, J. Radioanal. Nucl. Chem. Art. 177 (1994) 365. [8] M. Jurado Vargas, F. FernaH ndez de Soto, A. MartmH n SaH nchez, Nucl. Instr. and Meth. A 364 (1995) 349. [9] M. Jurado Vargas, F. FernaH ndez de Soto, Nucl. Instr. and Meth. A 368 (1996) 488. [10] C.W. Sill, Nucl. Chem. Waste Mgmt. 7 (1987) 239. [11] K. Megumi, T. Mamuro, J. Geophys. Res. 79 (1974) 3357. [12] A.B. Tanner, in: T.F. Gesell, W.M. Lowder (Eds.), Natural Radiation Environment III, Vol. 1, 1980. [13] G.J. Hanckock, P. Martin, Appl. Radiat. Isotopes 47 (1996) 131. [14] M. Jurado Vargas, F. FernaH ndez de Soto, Appl. Radiat. Isotopes 47 (1996) 129.
A model to explain simultaneously the Rn and Rn emanation from thin electrodeposited sources M. Jurado Vargas* Departamento de Fn& sica, Universidad de Extremadura, 06071 Badajoz, Spain Received 13 July 1999; received in revised form 5 November 1999; accepted 29 November 1999
Abstract In thin radioactive sources, loss of radon by emanation is a very common phenomenon, especially in sources made by electrodeposition. A quanti"cation of this e!ect in radium sources can be easily developed by using a simple model that assumes a radon di!usion term in the ingrowth equations. By measuring the corresponding Rn/Ra activity ratio, a constant di!usion factor can be determined which represents the Rn emanation from the whole source. However, this simple model cannot explain simultaneously the Rn and Rn di!usion produced in a thin source, because it gives di!usion factors that are di!erent by many orders of magnitude for these two isotopes, while these values must be fairly close. In this paper, a new model of di!usion is proposed, which includes a linear dependence of the di!usion factor on the depth of Rn nuclides in the source. This new model has been applied to radium electrodeposited sources and allows us to explain satisfactorily both the Rn/Ra and Rn/Ra activity ratios observed in thin sources. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Several techniques have been described in the literature for the determination of radium in environmental samples. These include gamma ray spectrometry, liquid scintillation counting, radon emanation and alpha particle spectrometry using the coprecipitation of radium with barium sulphate [1,2]. Although this last technique has given satisfactory results for radium determination, since the losses of radon by di!usion from the source are then minimal mainly due to the thickness of the Ba(Ra)SO layer, it o!ers poor energy resolutions. This problem has been solved using procedures for
* Tel.: #34-24-289300; fax: #34-24-275428. E-mail address: [email protected] (M.J. Vargas).
radium electrodeposition [3}7], which allow one to achieve thinner layers and improved spectra with better energy resolutions, but also involve losses of radon from the sources. In previous work, we have performed some studies about Rn di!usion in Ra sources made by electrodeposition [8,9]. The loss of radon by emanation was observed by measuring the experimental Rn/Ra activity ratio versus the time elapsed from the Ra source preparation, and a quanti"cation of this phenomenon was developed by using a simple model that assumes a radon di!usion term in the ingrowth di!erential equations. In this way, a constant radon di!usion coef"cient D was deduced for each Ra source. These values ranged approximately between 0.10 and 0.23 days\ for a great number of Ra electrodeposited sources.
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 2 8 9 - 9
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
In spite of the good behaviour of this simple model in explaining the Rn di!usion, it however cannot explain simultaneously the Rn and Rn di!usion produced in a thin source, because it gives di!usion factors that are di!erent by many orders of magnitude for these two isotopes, while these values must be fairly close. In this paper, a more realistic model of di!usion is proposed, which includes a linear dependence of the di!usion factor on the depth of Rn nuclides in the source. This new model has been applied to radium electrodeposited sources and allows us to explain satisfactorily both Rn/Ra and Rn/Ra activity ratios observed in thin sources.
2. Constant di4usion model In order to describe the radon emanation e!ect, the simple model introduces a di!usion term into the ingrowth di!erential equations. This term represents the radon emanation rate and is proportional to the number of radon nuclides present at each time, with a constant di!usion factor D. Hence, the corresponding di!erential equation can be written as dN(Rn)"j(Ra)N(Ra) dt![j(Rn)#D]N(Rn)dt. (1) By solving this di!erential equation, one "nds that the activity ratio Rn/Ra at each time is given by
Rn j(Rn) (t)" Ra j(Rn)!j(Ra)#D) ;[1!exp([j(Rn)!j(Ra)#D]t)]. (2)
This expresion may be applied to both the Rn/Ra or Rn/Ra activity ratios. If we take into account the decay constants k of Rn, Ra, Rn and Ra (2.098;10\; 1.374;10\; 0.01247 and 2.192;10\ s\, respectively), the exponential term in Eq. (2) will be negligible for su$ciently large times after the radium source preparation, so that the Rn/Ra and Rn/Ra activity ratios will
609
reach a constant value given by
j(Rn) Rn " . Ra j(Rn)!j(Ra)#D
(3)
The Rn/Ra ratio will reach the constant value given by expression (3) at a time of about 20 days after the source preparation, while the Rn/Ra ratio will reach its constant value in a far shorter time of the order of a few minutes, due to the much smaller half-lives of Rn and Ra. In other words, the exponential term in the Rn/Ra ratio does not fall to zero immediately as e!ectively occurs for the Rn/Ra ratio. In previous work [8,9] we have observed a significant Rn emanation from Ra sources prepared by using some radium electrodeposition methods [5,6], as seen in the alpha-particle spectrum of electrodeposited Ra shown in Fig. 1 for example. Using the model of constant di!usion, the experimental Rn/Ra activity ratios were "tted versus time to the theoretical Eq. (2). Table 1 lists some typical values found for the Rn/Ra activity ratio, which range from 0.41 to 0.56, and the values of the constant di!usion factor D calculated for each source, which range typically between 0.142 and 0.261 days\ (see third column in Table 1). Also, several authors have reported signi"cant Rn retentions in thin radium sources and samples of soils and rocks [10}13], so that Rn di!usion does not allow the presumption of secular equilibrium of Rn with its parent Ra. In particular, Hanckock and Martin [13] observed typical Rn retentions that ranged from 0.57 to 0.77 for radium electrodeposited sources by using the same method as ours. Sill [10] noted that the Rn retention of a thin radium source prepared from a "ne precipitate of barium sulphate was 58%. Taking into consideration that the di!usion coef"cient D for Rn has to be fairly close to that of Rn, we can calculate the theoretical values of Rn retention that would be obtained in the source, by using as di!usion coe$cient in equation (3) with the value obtained for Rn. These values for Rn retentions are listed in the fourth column of Table 1. As seen in this table, these theoretical values con"rm the theoretical secular equilibrium
610
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
Fig. 1. Alpha-particle spectrum of Ra obtained from a standard solution by using the electrodeposition procedure proposed by Whitehead et al. [6]. Note the high degree of disequilibrium between Rn and Ra, indicating a major di!usion of Rn from the source. Table 1 Experimental Rn/Ra activity ratios observed by the author in some radium electrodeposited sources (after 30 days following the electrodeposition) and the values of the Rn constant di!usion coe$cient D calculated for each Ra source (third column). The fourth column gives the theoretical value of the Rn/Ra activity ratio that would be obtained in the source by assuming the calculated Rn di!usion coe$cient for Rn in Eq. (3). Numbers in parentheses denote the reference for the electrodeposition procedure Electrodeposition Observed procedure Rn/Ra
Calculated D [days\]
Calculated Rn/Ra
[6] [5] [6] [6] [5] [5] [6] [5],[6] [5]
0.261 0.250 0.240 0.213 0.204 0.196 0.167 0.148 0.142
0.999758 0.999768 0.999777 0.999802 0.999811 0.999818 0.999845 0.999863 0.999868
0.41 0.42 0.43 0.46 0.47 0.48 0.52 0.55 0.56
between Rn and Ra. The reason for this result is that, although Rn (half-life"3.8 days) is di!used by emanation from the source, the di!usion of Rn must be practically null due to its short half-life (3.96 s). In other words, for this last nuclide, the decay rate would be much higher than any possible emanation rate, so that there will be negligible Rn emerging from the source. Obviously, these results are not in agreement with the experimental values found in the literature for radium electrodeposited sources, which involve signi"cant Rn retentions. We have already reported this disagreement in the constant di!usion model in a previous paper [14].
3. Linear di4usion model A new model is proposed. Firstly, it is assumed that Ra and Rn nuclides in the source are uniformly
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
611
Although the mean Rn di!usion length is high in soils and other materials, there are evidences for a practically null di!usion length in some thin sources, where the Rn losses are minimal [9]. These results are in accordance with the existence of a null di!usion zone, as established here. The Rn di!usion coe$cient will then have the following values:
D(x)" Fig. 2. Simple distribution model of Ra and Rn nuclides in the source. Nuclei are deposited uniformly to a thickness L in the source.
distributed, so that nuclides are `incrusteda homogeneously to a depth `¸a into the source, as shown schematically in Fig. 2. In this way, a constant distribution function for Rn nuclides is obtained, which normalized is given by F"1/¸. Secondly, the constant Rn di!usion coe$cient introduced in the simple model must be substituted by a new, more realistic coe$cient that exhibts a dependence on the depth of Rn nuclides in the source. The situation is illustrated in Fig. 3, where Rn atoms that are deeper in the source (zone from 0 to a¸) are assumed not to escape via di!ussion, while the rest undergo linear di!usion potential reaching a maximum value D for the atoms located at the surface of the source.
Fig. 3. Schematic illustration of the linear di!usion model. Rn nuclides that are deeper in the source do not di!use, while the rest undergo linear di!usion relative to the location of the Rn nuclides in the source.
0
aD D x! ¸(1!a) (1!a)
for 0(x4a¸, for a¸4x4¸.
(4)
The Rn/Ra activity ratio must then be given by the contributions of all the nuclei located at di!erent depths in the source, which have di!erent values of the di!usion coe$cient. Taking into consideration (3) for a constant di!usion coe$cient D(x), the Rn/Ra activity ratio in this linear model will be given by
* [Rn/Ra(x)]F(x) dx * j(Rn) " dx. (5) [j(Rn!j(Ra)#D(x)]¸ If we take into account that the radioactive constants j(Ra) and j(Ra) are several orders of magnitude smaller than j(Rn) and j(Rn), respectively, integration of Eq. (6) will give (Rn/Ra)"
(1!a) j(Rn) D (Rn/Ra)"a# In 1# . (6) j(Rn) D It must be noted that Eq. (6) can be applied to both Rn/Ra and Rn/Ra activity ratios. The proposed linear di!usion model gives a Rn/Ra activity ratio that depends only on two parameters, a and D . The "rst parameter, a corres ponds to the percentage of Rn nuclides in the source that do not undergo any di!usion due to their location at the deepest layers of deposition. The second parameter, D , corresponds to the di!u sion coe$cient for Rn nuclides located on the surface of the source, representing the maximum value of Rn di!usion. These two parameters are only related to the di!usion phenomenon and therefore have to be common to both Rn and Rn isotopes in a given source.
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M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
Table 2 Values obtained for the a parameter and values deduced for the Rn/Ra activity ratio by using the proposed linear di!usion model. Three typical values for the experimental Rn/Ra activity ratio obtained in radium electrodeposited sources and several values of the surface di!usion coe$cient D have been taken Assumed values
Calculated values
D (days\)
Rn/Ra
a
Rn/Ra
1
0.40 0.48 0.56
0.0896 0.2110 0.3324
0.999578 0.999634 0.999690
10
0.40 0.48 0.56
0.3525 0.4388 0.5252
0.997013 0.997411 0.997810
50
0.40 0.48 0.56
0.3875 0.4691 0.5508
0.986210 0.988047 0.989887
100
0.40 0.48 0.56
0.3930 0.4739 0.5549
0.973456 0.976994 0.980536
1000
0.40 0.48 0.56
0.3991 0.4792 0.5593
0.824146 0.847588 0.871029
5000
0.40 0.48 0.56
0.3998 0.4798 0.5598
0.623524 0.673704 0.723884
10 000
0.40 0.48 0.56
0.3999 0.4799 0.5599
0.500293 0.610468 0.670385
This linear model then has to explain both Rn and Rn retentions in a thin source. In order to check the model, several values for the Rn/Ra activity ratio and the surface di!usion coe$cient D were assumed (see Table 2). Three typical values of Rn/Ra were assumed, in agreement with those reported above in this present work in Table 1. From these values for the Rn/Ra ratio and the D coe$cient, the values for the a parameter were obtained, and hence the theoretical Rn/Ra activity ratios deduced. As seen in Table 2, the values obtained for the parameter a become very close to the Rn/Ra
ratio as the value of D increases. This means that, when a high rate of di!usion is involved, the Rn/Ra ratio observed in thin sources is essentially due to the percentage of Rn nuclides that do not di!use in the source (which is quanti"ed in the parameter a), while the rest of the Rn nuclides practically do not contribute to Rn retention due to their great degree of di!usion from the source. In other words, for these more super"cial Rn nuclides the emanation rate is very much greater than the decay rate; i.e., D is very much greater than j(Rn). In addition, an observation of Table 2 shows that high Rn retentions can only be obtained by assuming high values for the surface di!usion coef"cient D . In particular, Rn/Ra ratios in the range observed by Hanckock and Martin [13] for thin electrodeposited sources (values that ranged from 0.569 to 0.771) can be obtained for the three assumed values of the Rn/Ra ratio by using surface coe$cients D greater than 1000 days\. In this case, then, the Rn/Ra activity ratio is due to the sum of two contributions: a "rst term given by the value of the parameter a, due to the Rn nuclides located in the deepest layers that do not undergo di!usion, while the second term is given by the more super"cial Rn nuclides, which also contribute to the Rn/Ra activity ratio, because the decay rate for Rn is signi"cant and cannot be hidden by the di!usion rate. In other words, here j(Rn) cannot be neglected versus D . In summary, as seen from Table 2, the linear di!usion model allows one to explain both Rn and Rn retentions in thin sources, by using the same function D(x) for the Rn di!usion coe$cient. The high Rn and Rn retentions observed in thin alpha sources can only be explained by assuming a steep slope in the D(x) function, i.e., very high values of the surface coe$cient D . In particular, for electrodeposited sources, this model establishes a parameter a given by the value of the Rn/Ra activity ratio and values of D of the order of 10}10 days\. Finally, an important point must be taken into consideration. The parameter a must be related to the fraction of Rn nuclides which di!use through the solid deposit and reach the sample pores. Therefore, it may be expected that the parameter
M.J. Vargas / Nuclear Instruments and Methods in Physics Research A 447 (2000) 608}613
a for Rn and Rn would be di!erent, due to the di!erences in their half-lives. A new model based on the present one must be elaborated to account for this aspect. 4. Conclusions The loss of Rn by emanation in thin radioactive sources can be easily quanti"ed by assuming a Rn emanation term in the ingrowth equations, with a constant di!usion coe$cient. However, this simple model cannot explain both Rn and Rn emanations simultaneously, due to the very di!erent half-lives of these isotopes. In this present paper, a new model has been proposed which assumes two di!erent zones in the radioactive source: a "rst zone very deep in the source where Rn di!usion is null, and a second one, less deep, with a linear dependence of the di!usion factor on the depth of Rn nuclides in the deposit. In particular, the percentage of Rn nuclides in the "rst zone of the source is given by the observed Rn/Ra activity ratio, while high values of the surface coe$cient D (about 10}10 days\) must be involved in thin sources to derive the observed Rn retentions. This new model allows one to explain satisfactorily the high Rn and Rn retentions observed in thin sources, using the same function for the di!u-
613
sion coe$cient for both isotopes Rn and Rn. Thus, this model gives a more satisfactory explanation of the Rn emanation phenomenon observed in thin alpha sources. In spite of this, new models based on the present model must be proposed to allow di!erent values of the parameter a for Rn and Rn.
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