Radon diffusion and the emanation fraction for NIST polyethylene
capsules containing radium solution
Peter Volkovitsky
National Institute of Standards and Technology, Gaithersburg MD 20899-0001
Ionizing Radiation Division, Building 245, C110, National Institute of Standards and
Technology, 100 Bureau Drive, Gaithersburg MD 20899-8462, USA
Fax 1 301 926 7416, E-mail: peter.volkovitsky@nist.gov

Abstract
The national standard for radon (222Rn) measurements in the United States is
based on Standard Reference Materials (SRMs) prepared at the National
Institute of Standards and Technology (NIST), Gaithersburg, MD. NIST
radon standards are hermetically sealed polyethylene capsules, filled with
radium (226Ra) solution. In preparation of standard reference materials for
222
Rn, it is necessary to understand precisely the emanation fraction of radon
from the internal radium solution, through the walls of the polyethylene
container, to the surroundings. In preparation of a new radon SRM, it was
found that the emanation fraction dependence on accumulation time is not
accurately described by the standard ingrowth factor. A mathematical
description of the emanation model is presented, which is in agreement with
the empirical results for NIST radon emanation standards. It is shown that
the radon accumulated inside the polyethylene walls of the capsule is
significant and measurable. The radon diffusion coefficient of polyethylene
is estimated to be 200 times less than the radon diffusion coefficient of
water, while radon solubility in polyethylene is estimated as approximately
2/3 of the solubility of radon in water. This approach could be applied to
describe the emanation of radon through other materials.
Keywords: radon, standards, emanation factor, radon diffusion

1

1. Introduction
The national standard for radon (222Rn) measurements in the United States is based on
Standard Reference Materials (SRMs) prepared at the National Institute of Standards and
Technology (NIST), Gaithersburg, MD. NIST radon standards are hermetically sealed
polyethylene capsules, filled with radium (226Ra) solution. The first production run of NIST
radon emanation SRMs was developed more than 10 years ago (Collé and Hutchinson,
1993).These standards are used widely as primary radon standards for calibration of radon
detectors and monitors (Collé at al., 1995, Kotrappa and Stieff, 1994). These early radon
SRMs (SRM 4968) were produced in 1993 in sealed polyethylene capsules with total radium
solution activities of approximately 5 Bq, 50 Bq, and 500 Bq. With the depletion of this
SRM stock, a new series of SRM was produced near the end of 2003. In this case, each
activity level was assigned a unique SRM identification (i.e., SRM 4971 for 5 Bq capsules,
SRM 4972 for 50 Bq capsules, and SRM 4973 for 500 Bq capsules). The overall dimensions
of the cylindrically shaped polyethylene capsules used for the new series are nominally 26
mm in length by 4.4 mm diameter, with an inner solution-containment capacity (vin ) of
approximately 0.180 cm3 (Figure 1).
The method of measurement involves physical separation of gaseous radon from the parent
radionuclide (by emanation of the gas through the walls of the containment capsule), and
quantitative transfer of the gas to ionization chambers for measurements. The emanation
fraction, the activity of 222Rn outside the capsule vs the activity of 226Ra solution confined
inside the capsule is the most important parameter of 222Rn standards. A complete
description of the gas purification and counting apparatus is presented elsewhere (Colle et al.,
1990). In this paper, we restrict our discussion to understanding the mechanism of radon
diffusion through capsule walls and to contribution of the polyethylene walls of the
containment capsule to the emanation of radon.
In the work presented here, the activity of confined radium solution ARa was measured at a
reference time 01/01/2004 12:00 EST with an uncertainty of 1.12% (coverage factor k = 2,
see Taylor and Kuyatt, 1994). The radium solution is contained in a polyethylene cylinder
(capsule) with the length of internal volume of 20 mm and an inside diameter of 3.4 mm
(Figure 1). When such a capsule is placed into a closed vial filled with air at almost 100%
humidity and normal temperature and pressure, and kept there for some accumulation time t,
radon diffuses through the polyethylene walls of the capsule into the vial, and the activity of
radon outside the capsule Aout can be written as Aout = f·ARa, where f is an emanation
fraction. In this paper, calculation of f as a function of accumulation time, t, is presented first
in a simple diffusion-based two-box model. The results of calculations are compared with
experimental data obtained in preparation of the new radon emanation standards and thus to
determine the emanation fraction and its uncertainty for new radon emanation standards. It is
found that radon accumulated inside the polyethylene walls of the capsule is significant and
measurable. A mathematical model is presented that is in agreement with the experimentally
determined radon activity. The approach to development of the revised model could be
applied to describe the emanation of radon through other materials.

2. Diffusion equations for the capsule
Consider the accumulation of radon in a glass vial of volume vout, where vout is much larger
than vin. Assume that radon both inside and outside of the capsule has a uniform

2

concentration (i.e., constant concentration throughout solution and air volumes). This
assumption is quite reasonable because the diffusion of radon in the air and in the radium
solution is a much faster process than the radon diffusion through polyethylene. The radon
diffusion coefficient of the air is Da = 0.1 cm2/s (NCRP, 1988), and the radon diffusion
coefficient of water is Dw = 105 cm2/s (Rona, 1917, Broecker and Peng, 1974). The radon
diffusion coefficient of polyethylene Dp has not been measured, however, based on the
results of experiments presented here, the value of Dp is more than two orders of magnitude
less than that of water.
Convection of radon in air and in water greatly decreases the time needed to achieve constant
concentration – both inside and outside the capsule. It will be shown that experimental
results clearly indicate that, with accumulation time, the amount of radon in the walls of the
polyethylene becomes a measurable quantity, which should be accounted for. Moreover, it
will be demonstrated that the fraction of radon inside the polyethylene of a capsule is not
only measurable, but also significant relative to radon accumulated inside the radium
solution.
To begin, we consider the basics of diffusion of gases. One-dimensional gas diffusion in
terms of flux J is described by Fick’s First Law, which states that the diffusive flux is directly
proportional to the concentration gradient:
J = −D

∂c
∂x

(1)

where J = 1 S ⋅ ∂N ∂t is the flux of radon through the area S, c = N v is the concentration
of N gas atoms in the volume v, x is the dimension along which flux takes place, and D is the
diffusion coefficient.

Under the above assumptions we can write two equations for the number of radon atoms
inside (Nin) and outside (Nout) of the capsule:
SD p ⎛ N in N out ⎞
dN in
⎜
⎟ − λN in ;
= ARa −
−
d ⎜⎝ vin
vout ⎟⎠
dt
dN out SD p ⎛ N in N out ⎞
⎜
⎟ − λN out .
=
−
dt
d ⎜⎝ vin
vout ⎟⎠

(a )
(2)
(b)

Here ARa is the total activity of radium solution in decays per second (Bq), i.e. the rate of
radon atom production inside the capsule and S is the side-surface area of the capsule
(neglecting the thick sealing plugs at each end). Decay of 226Ra (half life ~ 1600 years) is
considered to be small for this discussion. With dimensions shown in Figure 1, S = 2.45 cm2.
d is the thickness of a side wall of the capsule, which is equal to 0.05 cm. vin and vout are
the volumes inside and outside the capsule. With dimensions in Figure 1, vin = 0.18 cm3.
The bubbler, where the capsule was embedded for radon extraction, has volume vout = 200
cm3. The ratio k = vin / vout = 9·104.
is the decay constant of radon, = 2.09822·106 s1
(ENSDF, 2004).

3

It is convenient for this discussion to simplify and rewrite these equations in terms of
activities by denoting μ = SDP vin d , the time constant of diffusion, and introducing the
activities of radon Ain = Nin and Aout = Nout. In this way, one obtains the following
equations:

dAin
= λARa − (λ + μ )Ain + kμAout ;
dt
dAout
= −(λ + kμ )Aout + μAin
dt

(a )
(3)
(b)

Adding these two equations, the standard equation for the total radon activity ARn = Ain +
Aout is obtained:
dARn
(4)
= λ ( ARa − ARn )
dt
s
with the stationary solution ARn
= ARa , and the general solution in the form:

ARn (t ) = ARa [1 − exp(−λt )] + ARn (0) exp(−λt )

(5)

here ARn(0) is the amount of radon inside the capsule before the bubbler was sealed and
diffusion started.
The stationary solutions of equations (3) are:

λ + kμ
λ
;
≈ ARa
λ+μ
λ + μ + kμ
μ
μ
= ARa
≈ ARa
λ + μ + kμ
λ+μ

Ains = ARa
s
out

A

(a)
(6)
(b)

Since the volume inside the capsule is small, the ratio of inside to outside volumes, k, is
always a small number (less than 10 ÷ 10 ). For the stationary solution, the emanation
fraction f 0 = μ (λ + μ ) . If radon diffusion is much faster than the radon decay ( >> ),
f0 = 1; in the opposite limit ( >> ), f0 = 0, and there will be no radon outside of the
capsule.
To measure f0 directly, the accumulation time of radon in the bubbler, t, should be much
larger than 1/( + ), which may be as great as a few weeks. Fortunately, when many
sequential measurements were to be accomplished (as is the case for the current work), by
modeling the time dependence of the emanation fraction f for finite time t, much shorter
accumulation periods could be used (on the order of days).
The general solution of equations (2) has the forms:

4

Ain (t ) = Ains + kC1 exp(−λt ) + C 2 exp(−(λ + μ + kμ )t );

(a)

+ C1 exp(−λt ) − C 2 exp(−(λ + μ + kμ )t )

(b)

Aout (t ) = A

s
out

(7)

At t = 0 there is no radon outside the capsule: Aout(0) = 0. Denote Ain(0) = A0. Constants
C1 and C2 can be determined from these initial conditions, and the solution (7 b) for Aout(t)
has the form:
Aout (t ) = ARa f 0 [1 − exp(−(λ + μ )t )] − ( ARa − A0 ) exp(−λt )[1 − exp(− μt )]

(8)

Before the capsule is loaded into the bubbler, it was kept in a closed vial with volume 20
cm3. The outside volume is still much bigger than the inside one, and k << 1. Under these
conditions A0 = ARa λ (λ + μ ) = ARa (1 − f 0 ) (see equation (6 b)). In this case, C2 = 0 and
equation (8) is reduced to:
Aout (t ) = ARa f 0 [1 − exp(−λt )]

(9)

and f ( x) x = Aout ( x) xARa = f 0 , where x = 1 − exp(−λt ) . Under this assumption the
emanation fraction f0 was determined in previous work (Collé and Hutchinson, 1993).

3. Results of measurements and modification of the model
The recent experimental data exhibits some dependence of f(x)/x on x (Figure 2). This
dependency suggested a measurable accumulation of radon in the polyethylene walls of the
container. Measurements were performed with the NIST Radon Pulse Ionization Chambers
facility described in Collé, 1990. Capsules were stored in closed glass vials with volume of
20 cm3, prior to use and then transferred into bubblers with volume of 200 cm3. Radon was
allowed to accumulate inside the bubbler for time interval t between 1 and 18 days. A total
of 62 measurements were performed —17 of which measured capsules that were
“preconditioned” according to the recommendations of Collé and Hutchinson, 1993.
Preconditioning of these 17 capsules consisted of interim storage of the capsules in a standard
fume hood for one day after removal from the glass vial and before being placed into the
bubbler for accumulation and counting. A summary of the modeled experimental results is
given in Table 1. No significant difference was observed between “preconditioned” and
“non-preconditioned” capsules. The infinitely large volume under the hood gives the ratio of
volumes, k = 0, while 20 cm3 volume of a glass vial corresponds to k = 0.01, which is
considered here as negligible. In accordance with the model, data shown in Figure 2 do not
reveal any noticeable difference between “preconditioned” and “non-preconditioned”
capsules.
A more detailed description of capsule production and measurement will be given elsewhere.
To explain the observed dependence of f ( x) x on x, the initial activity of radon inside the
capsule should be somewhat bigger than A0 = ARa (1 − f 0 ) , namely A0 = ARa (1 − f 0 + α ) .
Constant may be considered as a fraction of radon inside of a polyethylene of a capsule.
Then:

5

Aout (t ) = ARa f 0 [1 − exp(−λt )] + αARa exp(−λt )[1 − exp(− μt )]

(10)

As it follows from Figure 2, constant f 0 ≈ 0.85 and μ ≈ 5.7λ . For t ≥ 1 λ , exp(− μt ) << 1 ,
and in terms of x = 1 − exp(−λt ) , introduced above, equation (10) can be rewritten as:
Aout (t ) = ARa f (t ) = ARa { f 0 [1 − exp(−λt )] + α exp(−λt )} = ARa [ f 0 x + α (1 − x )]
and

f ( x) = f 0 x + α (1 − x )

(11)

(12)

4. Experimental data fits and parameter determination

In Figure 3 the fit of all experimental data with formula (12) is shown. The fit gives
f ( x) = (82.14 x + 5.09)% with f 0 = (87.23 ± 0.72)%, α = (5.09 ± 1.20)% Uncertainties in
parameters correspond to two standard deviations of regression parameters (see, for example,
Draper and Smith, 1981). The total fraction of radon both dissolved in the radium solution
inside the capsule and inside the polyethylene walls of the capsule is equal to 1 − f 0 = 0.128 .
Given = 0.051, this means that the radium solution contains 1 − f 0 − α = 0.077 of total
radon produced, and the polyethylene walls contain approximately 2/3 of the radon dissolved
in the radium solution inside the capsule. Based on this fit, one can determine the
μ = 6.85λ = 1.44 ⋅ 10 −5 s −1 , and the radon diffusion coefficient of polyethylene
D p = μdvin S = 5.2 ⋅ 10 −8 cm 2 s −1 , which is approximately 200 times less than the radon
diffusion coefficient of water.
Figures 4, 5, and 6 show fits with formula (12) for each SRM individually.
The obtained results are summarized in Table 1. All emanation fractions f0 coincide on the
level of three sigma uncertainties, however on the level of two sigma uncertainties the
preconditioned 50 Bq capsules give f0 less than others. Parameters determined for all
measurements coincide on the level of two sigma uncertainties.

5. Conclusions and future experiments

The emanation factor f0, obtained in previous measurements with polyethylene capsules
(Collé and Hutchinson, 1993) was f 0 = (89.0 ± 3.6)% . Based on the results of our present
measurements, the value of f 0 = (87.23 ± 0.72)% (coverage factor k = 2 for both
uncertainties), which is in agreement with the previous result, given the stated uncertainties.
However, while Collé and Hutchinson observed no noticeable dependence of f(x)/x on x, the
results presented here show a dependence of f(x)/x on x similar to that observed by Dean and
Kolkowski, 2004 for measurements of radon accumulation from polyethylene capsules in a
radon-in-water generator. These experimenters observed an f(x)/x dependence on x in
independent sets of experimental measurements of radon diffusion in data collected in 1996
and 2003. Although the radon diffusion coefficient in water is much less than that of air, the
value is much bigger than the radon diffusion coefficient of polyethylene. Thus, Eq. (12),
presented in this paper, should be valid for accumulation in a radon-in-water generator.
6

Measurements of emanation factor in the NIST radon-in water generator (see Collé and
Kishore, 1997) are planned for future work.
One of the results of the present work is an estimate for the radon diffusion coefficient of
polyethylene D p = μdvin S = 5.2 ⋅ 10 −8 cm 2 s −1 by measurement of emanation coefficient. A
method similar to the one of the present work may be used for other materials to measure the
radon diffusion coefficient.

Acknowledgements

The author is indebted to Ron Collé, Dan Golas, Robin Hutchinson, Larry Lucas, and Mike
Unterweger for discussions and to Andrew Rukhin for explanation of uncertainties of the
regression fit parameters. I would like to express my sincere gratitude to Mike Schultz for
careful reading of the manuscript and many valuable remarks.

References

Broecker, W.S., Peng T.H. 1974, Gas Exchange Rates between Air and Sea, Tellus 26, pp
21-35;
Collé, R., Hutchinson J.M.R., and Unterweger M.P., 1990, The NIST Primary Radon-222
Measurement System, J. Res. Natl. Inst. Stand. Technol., 95, pp 155-165;
Collé, R. and Hutchinson, J.M.R., 1993, Technical Notes on the Use of Standard Reference
Material. Radon-222 Emanation Standard, NIST Technical Report 4968;
Collé, R., Kotrappa, P., and Hutchinson, J.M.R., 1995, Calibration of Electret-Based Integral
Radon Monitors Using NIST Polyethylene-Encapsulated 226Ra/222Rn Emanation (PEPE)
Standards, J. Res. Natl. Inst. Stand. Technol., 100, pp 629-639;
Collé R. and Kishore, K., 1997, An update of the NIST radon-in-water standard generator:
its performance efficacy and long-term stability, Nucl. Instr. Meth., A 391, pp 511-528;
Dean, J.C.J. and Kolkowski, P., 2004, The development of a 222Rn Standard Solution
Dispenser at NPL, Applied Radiation and Isotopes 61, pp 95-100;
Draper, N.R. and Smith H., 1981, Applied Regression Analysis, John Wiley & Sons, 2nd Ed.;
ENSDF , 2004, Evaluated Nuclear Structure Data File, http://ie.lbl.gov/ensdf/welcome.htm;
Kotrappa P. and Stieff L.R., 1994, Application of NIST 222Rn Emanation Standards for
Calibrating 222Rn Monitors. Radiation Protection Dosimetry, 55, pp 211-218;
NCRP 1988, Measurement of Radon and Radon Daughters in Air, Report No 97;
Rona, E., 1917, Diffusiongrosse und Atomdurchmesser der Radiumemanation, Zeitschrift fur
Physikalische Chemie 92, pp 213-218;

7

Taylor B.N. and Kuyatt C. E., 1994, Guidelines for Evaluating and Expressing the Uncertainty
of NIST Measurement Results, NIST Technical Note 1297.

8

Tables

Capsule type
All
5 Bq only
50 Bq only
500 Bq only
All
5 Bq only
50 Bq only
500 Bq only
All capsules
5 Bq only
50 Bq only
500 Bq only

f0 (%)
All capsules
62
87.23 ± 0.36
19
88.39 ± 0.58
20
86.04 ± 0.47
23
86.98 ± 0.47
Preconditioned capsules
17
87.36 ± 0.73
5
89.18 ± 0.70
6
84.75 ± 0.30
6
87.25 ± 0.87
Non-preconditioned capsules
45
87.14 ± 0.46
14
87.68 ± 0.86
14
86.50 ± 0.64
17
87.36 ± 0.66

Number

α (%)
5.09 ± 0.60
6.38 ± 1.20
3.86 ± 0.78
5.64 ± 0.68
4.99 ± 0.92
6.13 ± 1.81
3.82 ± 0.37
6.09 ± 0.67
5.24 ± 0.89
7.23 ± 1.62
3.87 ± 1.27
4.41 ± 1.29

Table 1. Results of experimental data fit. Uncertainties correspond to one sigma level.

9

Figures

Figure 1.

Dimensions of the polyethylene capsule used for preparation of NIST radon
standard reference materials.

125
120
115

f(x)/x, %

110
105
100
95
90
85
80
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x
non-preconditioned

Figure 2.

10

preconditioned

0.9

1.0

90
80
70

f(x), %

60
50
40
30
20
10
0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.8

0.9

1.0

x
non-preconditioned

preconditioned

Figure 3
90
80
70

f(x), %

60
50
40
30
20
10
0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x
non-preconditioned

preconditioned

Figure 4

11

90
80
70

f(x), %

60
50
40
30
20
10
0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.8

0.9

1.0

x
non-preconditioned

preconditioned

Figure 5.
90
80
70

f(x), %

60
50
40
30
20
10
0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x
non-preconditioned

Figure 6.

12

preconditioned

Figure captions
Figure 1. The polyethylene capsule with radium solution inside.
Figure 2. Dependence of emanation factor f on accumulation time t for all measured capsules;

x = 1 − exp(−λ Rn t )

Figure 3. Fit for emanation fraction in the form f ( x ) = f 0 x + α (1 − x ) , where x = 1 − exp(−λ Rn t )
for all capsules. f 0 = (87.23 ± 0.72)%, α = (5.09 ± 1.20)% Coverage factor for uncertainty k =
2.
Figure 4. Fit for emanation fraction in the form f ( x ) = f 0 x + α (1 − x ) , where x = 1 − exp(−λ Rn t )
for all 500 Bq capsules. f 0 = (86.98 ± 0.94)%, α = (5.64 ± 1.36)% Coverage factor for
uncertainty k = 2.
Figure 5. Fit for emanation fraction in the form f ( x ) = f 0 x + α (1 − x ) , where x = 1 − exp(−λ Rn t )
for all 50 Bq capsules. f 0 = (86.04 ± 0.94)%, α = (3.86 ± 1.56)% Coverage factor for
uncertainty k = 2.
Figure 6. Fit for emanation fraction in the form f ( x ) = f 0 x + α (1 − x ) , where x = 1 − exp(−λ Rn t )
for all 5 Bq capsules. f 0 = (88.39 ± 1.18)%, α = (6.38 ± 2.40)% Coverage factor for uncertainty
k = 2.

13