annex h
TRANSCRIPT
-
ANNEXE H. ABSORBED FRACTIONS FOR ALPHA, ELECTRON, AND BETA EMISSIONS
A. C. James,* G. Akabani, A. BirchalLt N. S. Jarvis,t J. K. Briant* and J. S. Durham* *Pacific North west Laboratory, USA
f National Radiological Protection Board, UK
H.l. Introduction
Contents
H.2. Implementation of Source and Target Configurations
H.3. Absorbed Fractions for Alpha Particles
H.4. Absorbed Fractions for Mono-energetic Electrons
H.5. Absorbed Fractions for p- and @ Particles
H.6. Algebraic Approximations
References
459
460
461
462
465
471
471
tl. 1. Introduction
(HI ) In this annexe, the methods used in this report to calculate absorbed fractions for sources of short-range radiations within the respiratory tract arc briefly described. The resulting values of AF(T-S) that are recommended for substitution in the dosimetric model are shown graphically as functions of radiation energy in Chapter g. Tables H.l-H.6 give the values of AF(T-S) calculated for discrete values of the mean energy of short-range particulate radiations of each type. Algebraic formulae derived to approximate these calculated absorbed fractions and to interpolate between them are given in Tables H.7-H.lO. These formulae are provided for ease of calculation to evaluate AF(T- S) adequately for any radionuclide that emits alpha particles, electrons, or beta particles (negatrons or positrons). Other approximations may also be appropriate.
(H2) The average fraction of energy absorbed in target tissue T per emission of radiation R in source S is given by:
AF(T-S), = E(T+S), E R
U-II)
where E(T-S)a is the average energy (in MeV) absorbed by the target tissue in T per emission of radiation R in source S, and E, is the energy (in MeV) of the radiation.
The work of Or A. C. James, Or G. Akabani. Or J. K. Briar& and Or J. S. Durham on this annex was supported by the US Department of Energy under Contract DE-ACO6-76RLG 1830 with Battelle Memorial Institute.
The work of Or A. Birchall and Or N. S. Jarvis on this anncxc was partially supportrd by the Commission of the European Communities under Comract Bf6-0347-Item 2. and also by the UK Health and Safety Executive.
459
-
460 THE REPORT OF A TASK GROUP OF COMMITTEE 2
(H3) All of the absorbed fractions required to evaluate the respiratory tract dosimetry model as a function of the emitted radiation energy were calculated by a three- dimensional, Monte Carlo radiation transport method. This provided substantially improved accuracy in the absorbed fractions calculated for alpha emissions at high energies compared to the approximation method used earlier by the Task Group (James er aL, 1991). The Monte Carlo radiation transport code used here also increased the accuracy of absorbed fractions calculated for electron emissions over that possible with the point-kernel method used earlier (James et al., 1991). by taking realistic account of multiple electron-scattering and the effects of air-tissue interfaces.
H.2. Implementation of Source and Target Configurations
(H4) Figure H.1 shows in cross-section the four different arrangements of cylindrical radiation source and target that apply to the various regions of the respiratory tract. For each case, Chapter 2 defines the radius of the inner surface of the airway, and the radial extent of the underlying source. target, and absorbing tissue layers.
A. Intcmal Source (airway sufhcC of mucus)
C. External Source (srquesterd)
Source Region
la Tqct Region
D. External Infinite Source (alveoli)
El Unit Dunsiry Ahsorbcr cl Low Density Ahsdw
Fig. H. I. Cross-sectional diagrams of source and target configurations used to calculate absorbed fractions for the rcspirntory tract dosimctry model.
-
ABSORBED FRACTIONS FOR ALPHA. ELECTRON AND BETA EMISSIONS 461 (H5) Figure H.lA represents the geometry used to calculate absorbed fractions for
radiation sources at the airway surface, i.e. AF(ET, - surface). or in a layer of fluid at the airway surface. i.e. AF(ET: -fast mucus), AF(BB-fast mucus), or AF(bb-fast mucus). To calculate the absorbed fractions AF( BB - slow mucus) and AF(bb - slow mucus), the layer of shielding at the airway surface provided by the overlying fast mucus was added.
(H6) Figure H.lB represents the arrangement of source. target, and shielding layers used to calculate absorbed fractions for radionuclides that are bound in the epithelial tissue, i.e. for AF(ET, - bound), AF(BB + bound), and AF(bb- bound), where the target layer comprises part of the source.
(H7) Figures H.lC and H.l D represent the two classes of source-target geometry where the source is entirely external to the target layer. Figure H.lC represents the geometrical arrangement used to calculate AF(ET2 + sequestered). AF(BB-sequestered), and AF( bb - sequestered), where the source is composed of a thin macrophage layer of unit-density tissue. Figure H.lD represents all cases where the source is assumed to be distributed through alveolar tissue. of density 0.2 g cme3, and infinite extent, i.e. AF(BB-AI) and AF(bb-Al).
(H8) In order to carry out the Monte Carlo calculations of AF(BB c Al) and AF(bb-Al). the infinite alveolar-interstitial source was replaced by a finite volume bounded by the radius, Rcsda+ R,,. where RcrJa is the continuous slowing down approximation to the maximum range of the radiation considered, and R,, is the outer radius of the airway wall. The appropriate values of Rcda were obtained for alpha particles from ICRU Report 49 (ICRU, 1993), and for electrons from ICRU Report 37 (ICRU, 1983). Under these conditions, the absorbed fractions for an infinite source were ohtaincd by normalizing the calculated absorbed fraction. AF*(T- AI), as follows:
AF(T- AI) = AF*(T- AI) d(&.,,+R,,)*-%I k
(H2)
whcrc AF*(T-AI) is the absorbed fraction calculated for the source uniformly distributed in a cylindrical volume of material of density 0.2 g cm--, extending between radii R,, and R,, + R,,,,, (in cm), and k is the proportional volume of the whole AI source that corresponds to unit length of target airways. For the reference adult male, the total length of target airways in the BB region is modelled as 183.5 cm, and that of the bb region is 7756 cm. Thus the values of the normalizing constant, k, are 5500 cmJ G 183.5 cm = 29.97 cm? for the BB region, and 5500 cm3 i 7756 cm = 0.709 cm? for the bb region.
(H9) Monte Carlo sampling of the points of origin and direction of transport of the emitted radiation was carried out uniformly within each source. For each calculation, the number of radiation transport cases followed was chosen to achieve a standard error in the calculated mean absorbed fraction, if possible, of less than 1%. The number of radiation transport histories required to achieve this precision depends rather critically on the spatial extent of the source, and the nature of the radiation (alpha particle or electron).
H.3. Absorbed Fractions for Alpha Particles
(H 10) The Monte Carlo code used to calculate absorbed fractions for alpha particles (developed by G. Akabani, Pacific Northwest Laboratory) incorporated the stopping
-
462 THE REPORT OF A TASK GROUP OF COMMITTEE 2
power and range values for liquid water and air given on pp. 256 and 213, respectively, of ICRU Report 49 (ICRU. 1993).
(HII) The trajectory of each alpha particle was followed in the code until all of its energy had been deposited, and the total amount of energy deposited in each target region from many alpha particles emitted from each source was determined. For internal, bound and sequestered sources (Figs H. 1 A-H.1 C). 200.000 alpha particle histories were followed. For AF(bb-AI). where the alpha particle source is in the alveolar-interstitial region and the target is bronchiolar secretory cells, up to 2.000.000 histories were followed.
(H 12) Table H.l gives the values of absorbed fraction calculated for alpha particles of energy between 2.0 and 11.0 MeV. The table also shows the cut-off energy for each source-target combination. which corresponds to the minimum distance between points in the source and target.
(H 13) For bound sources. there is no cut-off energy. At low energies, the absorbed fraction approaches asymptotically a constant value. This maximum value is determined by the volumetric proportion of the source that is occupied by the target.
(H 14) For alpha particle sources in the AI region with the bronchial basal or secretory cells as target. the absorbed fractions AF(BB,,,-AI) and AF(BB,,,-AI) are always zero. because the thickness of absorbing tissue between the source and target exceeds the alpha-particle range.
11.4. Absorbed Eractions for Mono-rnergetic Electrons
(1 I 15) The Monte Carlo ccdc used to calculate absorhcd fractions for mono-encrgctic clcctrons (also dcvclopcd by G. Akabani, Pacific Northwest Laboratory) incorporated the tilcctron G;unma Shower transport code EGS4 (N&on cr al., 1985; Biclajcw CI al., I19 I ). The materials used in the code wcrc air, and water to simulate tissue.
(11 IO) The EGSJ code accurately represents radiation transport phenomena for electrons and photons down to I kcV. The code models the production of both knock-on clcctrons and bremsstrahlung above a certain energy threshold (taken to be 1 keV for these calculations). Transport of the electrons themselves is governed in the code by the Multiple Scattering theory. For these calculations. a practical upper Limit for energy loss in each scattering event was set at 6% of the current electron energy, i.e. the variable ESTEPE in the EGS4 code was set at 0.06. This value of ESTEPE is consistent with the small linear dimensions of tissue targets in which electron energy loss is to be followed. and allows accurate simulation of the electrons curved path.
(H 17) The energy deposited along an electrons path was scored in each cylindrical shell of absorber for each of the source-target arrangements shown in Figs H.lA-H.lD. Electrons and photons were transported until their energy dropped to I keV, which was assumed to be deposited locally. Thus, the history of all secondary electrons and photons (brcmsstrahlung) produced by each electron emitted in the source was followed completely.
(H IK) For internal, bound and sequestered sources (Figs H. I A-H. 1 C), the histories of 100.000 electrons of a given energy emitted in the source were followed. For AF(BB - Al) and AF(bb- Al). where the electron source is in the alveolar-interstitial region, up to 4.000,OOO histories were followed for electrons emitted with high energy.
-
Tabl
e H.
I.
\due
s of
abs
orbe
d fra
ctio
n,
AFtT
- S)
,, fo
r al
pha
parti
cles
T=wQ
ET
I ET
z n;
ET
2 W
.. Be
, B&
a.,
Be,
%a,
,
sour
ce(s
) sl
nfrc
sl
ufax
Bo
und
Sq-
Fast
Sl
ow
Boun
d se
qlBa
sIud
Al
M
ucus
M
ucus
2.00
0
0 0.
179
0.01
31
0 0
0.25
0 0.
0000
35
0
2.50
0
0 0.
175
0.03
74
0 0
0.25
1 0.
0022
8 0
3.00
0
0 0.
167
0.07
39
0 0
0.24
9 0.
0140
0
3.50
0
0 0.
161
0.11
2 0
0 0.
244
0.03
93
0
4.00
0
0 0.
153
0.13
5 0
0 0.
239
0.07
47
0
4.19
0
0 0.
151
0.13
9 0
0 0.
235
0.08
85
0
4.39
0
0 0.
147
0.14
1 0
0 0.
232
0.10
2 0
4.50
0
0 0.
147
0.14
2 0
0 0.
230
0.10
9 0
4.76
0
0 0.
143
0.14
2 0
0 0.
227
0.12
3 0
5.00
0
0 0.
140
0.14
1 0
0.00
0088
0.
221
0.13
2 0
5.15
0
0 0.
139
0.13
8 0
0.00
054
0.21
9 0.
135
0
5.50
0.
0012
7 O
X@00
76
0.13
5 0.
135
0.00
0063
0.
0050
6 0.
213
0.14
1 0
6.00
0.
0162
0.
0025
1 0.
131
0.12
8 0.
0050
6 0.
0217
0.
202
0.14
1 0
6.50
0.
0478
0.
0112
0.
125
0.12
1 0.
0218
0.
0435
0.
191
0.13
9 0
7.00
0.
0718
0.
0234
0.
118
0.11
4 0.
0509
0.
0600
0.
182
0.13
4 0
7.50
0.
0845
0.
0321
0.
111
0.10
7 0.
0802
0.
0778
0.
172
0.12
8 0
7.69
0.
0873
0.
0345
0.
109
0.10
4 0.
0893
0.
0857
0.
167
0.12
7 0
8.00
0.
0910
0.
0368
0.
104
0.10
1 0.
101
0.09
88
0.16
3 0.
123
0
8.50
0.
093 1
0.
0393
0.
0984
0.
0948
0.
112
0.11
0 0.
156
0.11
7 0
8.78
0.
0939
0.
0399
0.
0949
0.
0926
0.
114
0.11
5 0.
154
0.11
4 0
9.00
0.
0942
0.
0415
0.
0926
0.
0897
0.
116
0.11
5 0.
150
0.11
2 0
10.0
0 0.
0917
0.
0578
0.
0835
0.
0802
0.
119
0.11
9 0.
144
0.10
2 0
11.0
0 0.
0871
0.
0647
0.
0797
0.
0716
0.
116
0.11
6 0.
140
0.09
30
0
cubo
ff -6
Y.
5.21
5.
21
- 0.
79
5.29
4.
76
- 1.
79
- M
eV
-
Tabl
e H.
I. (
co~r
rincr
uc/)
Targ
el
BB,
BB,
BB,
BB,
BB,
bb,
bb,
bb,
bhc
s0ur
c.e
(S)
Fast
SI
OW
Bo
und
Sequ
estrr
ed
AI
Fnzt
Sl
ow
Boun
d Se
ques
tere
d AI
M
ucus
M
ucus
M
lhm
s M
IKXI
S
Ener
gy.
MeV
2.00
0
0 0.
500
0 0
0.00
526
0.05
62
0.38
7 0.
0020
9 0
2.50
0
0.00
388
O.b
99
0 0
0.03
87
0.11
0 0.
367
0.01
81
0 3.
00
0.00
022
0.02
27
0.49
7 0
0 0.
115
0.16
2 0.
338
0.05
44
0 3.
50
0.00
775
0.05
99
0.49
3 o.
ooo1
I 0
0.19
2 0.
203
0.30
5 0.
0918
0
4.00
0.
0327
0.
101
O.b
85
0.00
373
0 0.
231
0.23
5 0.
278
0.11
1 1.
22 n
lOA
4.19
0.
0455
0.
116
0.48
1 0.
0075
9 0
0.23
7 0.
241
0.27
0 0.
113
4.23
x I
O4
4.39
0.
0609
0.
132
0.47
7 0.
0136
0
0.23
9 0.
243
0.26
3 0.
114
1.07
x I
O
4.50
0.
0708
0.
140
0.47
4 0.
0182
0
0.24
0 0.
245
0.26
1 0.
115
1.61
x I
O
4.76
0.
0965
0.
159
0.46
7 0.
0304
0
0.23
8 0.
243
0.25
3 0.
114
3.40
x l
o-
5.00
0.
125
0.17
9 0.
459
0.04
32
0 0.
236
0.24
0 0.
247
0.11
3 5.
55 x
Iu
5.
15
0.14
4 0.
192
0.45
4 0.
0515
0
0.23
3 0.
237
0.24
4 0.
111
7.06
x 1
0
5.50
0.
189
0.22
7 0.
439
0.07
26
0 0.
225
0.22
9 0.
239
0.11
0 1.
08 x
IO
4 6.
00
0.24
9 0.
272
0.42
0 0.
102
0 0.
214
0.21
7 0.
227
0.11
7 1.
63 x
IO
4 6.
50
0.30
1 0.
309
0.39
8 0.
130
0 0.
200
0.20
2 0.
214
0.12
3 2.
17 x
IO
- 7.
00
0.33
4 0.
333
0.38
1 0.
14%
0
0.18
8 0.
190
0.20
2 0.
124
2.76
x I
O4
7.50
0.
348
0.35
0 0.
370
0.15
6 0
0.17
7 0.
178
0.18
8 0.
121
3.47
x I
O4
7.69
0.
353
0.35
5 0.
362
0.15
9 0
0.17
2 0.
173
0.18
3 0.
120
3.76
x I
O4
8.00
0.
353
0.35
6 0.
354
0.16
0 0
0.16
6 0.
166
0.17
5 0.
118
4.23
x I
O4
8.50
0.
349
0.35
4 0.
346
0.15
9 0
0.15
5 0.
155
0.16
5 0.
113
4.99
x 1
04
8.78
0.
346
0.35
0 0.
341
0.15
9 0
0.15
0 0.
151
0.16
0 0.
111
5.42
x l
Oa
9.00
0.
342
0.34
5 0.
339
0.15
7 0
0.14
6 0.
146
0.15
4 0.
108
5.71
x I
O-
10.0
0 0.
323
0.32
5 0.
324
0.15
5 0
0.12
8 0.
128
0.13
5 0.
0990
7.
04 *
lO
A 11
.00
0.30
0 0.
303
0.30
7 0.
167
0 0.
113
0.11
3 0.
121
0.08
94
8.23
x l
OA
cuio
ff Em
rIm.
2.70
1.
79
- 3.
20
- 1.
43
0.57
-
1.43
3.
56
-
ABSORBED FRACTIONS FOR ALPHA, ELECTRON AND BETA EMlSSlONS 365
(HI91 Table H.2 gives the values of absorbed fraction calculated for mono-energetic electrons of energy between 0.01 and 9.0 McV. For each source-target combination. the absorbed fractions were calculated for at least l-l, and generally more than 20. values of electron energy. Additional values shown in Table H.2 were obtained by cubic-spline interpolation between the calculated nodes.
(H20l In contrast to the transport of alpha particles, where the highest range in unit density tissue that needs to be considered is about 130 pm (for an I I-XleV alpha particle), high-energy electrons travel many centimetres. The csda range for a 9-MeV electron is about 4.5 cm in unit density tissue. and up to 22.5 cm in alveolated tissue. Therefore. a high-energy electron emitted in the wall of one airway can traverse many other airways in its path through the AI region.
(H21) However, the Monte Carlo calculation of absorbed fractions for all sources associated with the airway internal surface or airway wall (Figs H.lA-H.10 represents only the local absorption of energy in a single airway, which is assumed to be surrounded by an infinite cylinder of AI tissue. Additional energy deposition in each regional target tissue that results from cross-irradiation of remote airways is not represented explicitly in the calculation. However, this additional contribution can be adequately approximated by adding the surrogate calculated value of AF(T- AI) to each calculated value of the locally ahsorhcd fraction, AF(T- SdlrueJ. The values of AF(BB- mucus), AF(BB - bound), AF(BB- sequestercdl, and of AF(hb - mucus). AF(bb+ bound), AF(hh -sequestcrcd) given in Tahlc HZ!, therefore includs the respectivecomponcnts from AF(BH-Al) and AF(hb-All.
(H22) This USC of AF(T* AI) 21s it surrogate for the additive effect of II local airway source also irradiating remotc airways at high clcctron cncrgics will. in fact, tend to ovcrestimatc the contribution of airway cross-fire within the IN and hb regions when these arc considcrcd as discrete sources and tnrgcts. In reality, thcrc will aIs0 bc cross- fire at high cncrgics bctwccn the bb and I1fI rgions. and rice wr.w. This additional cross- fire will tend to off-set the ovcrcstimation of ahsorbcd fractions at high cncrgics that results from using AF(T- AI) to rcprcscnt rcmotc airway sources within each region. Consequently, it is not ncccssary to add in explicitly the rclativcly small fractions of emitted clcctron cncrgy absorbed by cross-fire bctwccn the BB and bb regions.
(H23) AS for alpha particles. Table H .2 also shows the cut-off electron energy for each source-target combination. This corresponds to the minimum distance bctwccn points in the source and target. Again, for bound sources. there is no cut-off energy. and at low energies the absorbed fraction approaches asymptotically the constant value determined by the volumetric proportion of the source that is occuped by the target.
H.5. Absorbed Fractions for @- and p Particles
(H24) Absorbed fractions for beta particles (negatrons and positrons) were calculated from the absorbed fraction data for mono-energetic electrons and the beta spectra for a series of radionuclidcs. The spectral-average absorbed fraction for a beta emitter is:
AF 8=
1 Y(E)EAF(T- S;E) dE
IY( dE (H3)
where Y(E) is the spectral yield at energy E.
-
Tabl
e H.
t. Va
lues
of a
bsor
bed
fract
ion,
AFi
T-
St,,
for
elec
trons
ii
sow
w
Sl
UfU
O
sluf
aeo
Boun
d tk
quda
d Fu
tMuc
ua
Slow
Mwu
a Bo
und
Sequ
este
red
Al
B z 2 >
0.01
0 0
0.01
5 0
0.02
0 0
0.02
5 0
0.03
0 0
0.03
5 0
0.04
0 0
0.04
5 0
0.05
0 4.
05 x
lo1
0.05
5 8.
65 x
IO
4 0.
060
8.13
x IO
* 0.
065
2.57
x lo
- 0.
070
4.84
K IO
J 0.
075
6.65
x I
O
0.00
0 8.
02 x
10
0.
085
8.91
x I
O
0 0 0 0 0 0 0 0 4.
42 x
lo-
7.59
x lo
1.
39 x
104
6.
09 x
lo-
1.34
x 1
OJ
2.20
x I
OJ
3.03
x 1
05
3.69
x I
O*
1.82
x IO
- 1.
82 x
lo
1.82
x IO
- 1.
80 x
IO
-1
1.75
x 1
0-l
1.66
x lo
1.
56 x
10
1.
49 x
IO-
1.43
x lo
1.
38 x
lo-
1.33
x lo
1.
27 x
lo-
1.20
x lO
- 1.
13 x
lo
1.07
x lo
1.
01 x
IO-
0 3.
69 x
10
8.
33 x
10.
1.
22 x
1O
J 4.
05 x
1O
J 8.
11 x
10
1.
19 x
lo-
1.42
x lo
1.
49 x
lo-
1.49
x 10
4 1.
42 x
lo-
1.34
x lo
- 1.
24 x
lo
1.17
x lo
1.
08 x
lo
1.01
x lo
-
0 0 0 0 0 0 0 0 2.
22 x
lo
2.06
x I
O4
3.03
x 1
04
1.36
x I
O=
3.16
x lO
a 5.
45 x
104
7.
76 x
lOa
9.62
x 1
OJ
0 0 0 0 0 0 0 1.
15 x
10
1.
31 x
lo4
2.53
x IO
* 1.
13 x
10-J
2.
62 x
10
4.
49 x
1O
J 6.
46 x
lo3
8.32
x 1
0
9.87
x 10
2.50
x lO
- 2.
50 x
IO
- 2.
50 x
lo-
2.50
x lo
2.
50 x
lo
2.47
x lo
2.
42 x
lo-
2.36
x lo
2.
28 x
16
2.18
x lo
2.
07 *
lo
1.96
x 1
0
1.86
x lo
1.
76 x
lo
1.67
x lo
1.
58 x
10-
l
0 0 0 3.
24 x
lo+
2.61
x 1
0.
1.69
x lO
J 4.
58 x
lo*
8.11
x 1
0
1.15
x lo
1.
35 x
lo
1.45
x lo
1.
49 x
lo-
1.45
x lo
1.
39 x
lo-
1.33
x lo
- 1.
27 x
10-
l
0 0
0 $
0 s
0 8
0 CI
0 s
0 0 3 m
0
N
-
0.09
0 9.
36 x
10'
0.
095
9.54
x 1
04
0.10
0 9.
57 x
1O
J 0.
1125
9.
04 x
IO
J 0.
125
8.40
~ lO
a 0.
150
7.03
x I
OJ
0.17
5 5.
90 x
!O
J 0.
200
5.12
x lo
= 0.
300
3.14
x 1
04
0.40
0 2.
19 x
IO
= 0.
500
1.66
x 10
3 0.
600
1.33
x 1
OJ
0.70
0 1.
10 x
105
0.80
0 9.
34 x
1O
J 0.
900
8.15
x I
O-'
!.ooo
7.
25 x
1O
J 1.
500
4.36
x I
O*
2.00
0 3.
07 K
IOJ
3.00
0 1.
93 x
10J
4.00
0 1.
41 x
lo=
9.00
0 5.
50 x
IO
4
4.26
x IO
' 4.
83 x
104
5.
36 x
10J
6.
19 x
IO*
6.36
x lO
a 5.
59 x
!O
J 4.
69 x
IO*
3.97
x !
OJ
2.25
x IO
* 1.
50 x
104
1.
12 x
loa
9.06
* 1
0.'
7.63
x IO
-' 6.
61 x
lo*
5.83
x 1
0"
5.22
x 1
0J
3.50
x !O
J 2.
67 x
lo-'
1.85
x lO
a I.4
2 *
lo-'
6.61
* 1
04
9.48
x 1
0'
9.00
x 1
0'
8.60
~ IO
* 7.
86 *
IO
* 7.
39 x
103
6.
78 x
IO
4 6.
07 x
IO*
5.30
x I
OJ
3.34
x 1
0-J
2.40
1.1
04
1.87
x IO
J 1.
54 x
104
1.
31 x
IO
J 1.
13 x
10-
z 9.
95 x
10'
8.
88 x
103
5.
72 x
IO-'
4.21
* 1
0.'
2.80
* I
O*
2.02
* l
o-'
8.69
* I
O4
9.36
x 1
0'
8.83
x lO
a 8.
32 x
lo*
7.11
x 1
0-J
6.38
x lo
* 5.
84 x
IO*
5.35
x 1
04
4.75
x 1
05
3.09
x 1O
J 2.
26 x
10'
1.
77 x
104
1.46
x 1
05
1.23
x IO
* 1.
07 x
104
9.50
* 1
04
8.54
x 1
05
5.78
x IO
* 4.
31 x
104
2.
90 *
10.
' 2.
15 *
10"
9.
65 x
IO
4
1.09
x lo
- 1.
16 x
lo"
1.19
x lo
-' I.1
9 *
lo"
1.13
x lo
- 9.
66 x
lOa
8.24
* I
O=
7.12
x lo
* 4.
37 x
!O
J 3.
07x
103
2.35
x IO
* 1.
89 x
lOa
1.56
x I
O<
1.32
* IO
3 1.
14 x
loa
9.86
x IO
-' 5.
65 x
105
4.
56 x
IO-'
3.08
* 1
0.'
2.39
* I
O-'
1.18
x 1
0.'
1.10
x lo
- 1.
17 x
!WL
1.20
x lo
-' 1.
20 x
lo"
1.14
x lo
-' 9.
71 x
1O
J 8.
27 x
10"
7.
15 x
IO
4 4.
39 x
10'
3.
09 x
104
2.
35 x
lOa
1.89
x lO
a 1.
56 x
IO*
1.32
x lO
a 1.
14 x
lo*
1.01
x 10
' 6.
11 x
lO
a 4.
58 x
lo=
3.08
* 10
" 2.
41 x
lO
a 1.
17 x
IO"
1.50
x lo
-' 1.
46 x
lo-'
1.43
x lo
-' 1.
36 x
lo-'
1.27
x lo
" 1.
09 x
lo"
9.32
x I
O'
8.01
x lO
a 4.
90x
lo=
3.42
x 1
0'
2.62
x 1
0J
2.12
x lo
* 1.
76 x
lO
a 1.
47 x
IO'
1.28
* lO
a I.1
3 x
IO4
7.10
x 1
0"
5.22
x I
O=
3.51
x 1
05
2.72
* 1
0.'
1.35
* IO
4
1.21
x lo
-' 1.
14 x
lo"
1.07
x lo
-' 9.
71 x
10'
9.
36 x
10'
8.
77 x
IO
= 7.
85 x
IO4
6.83
x IO
* 4.
34 x
lo=
3.13
x I
O4
2.37
x lO
a 1.
90 x
103
1.
58 x
104
1.
34 x
lo=
I.15
x 10
J 1.
02 x
!O
J 6.
44 x
lo-'
4.78
x IO
-' 3.
28 x
10"
2.
54 *
IO
5 1.
30 x
104
0 0 0 0 0 0 0 0 1.
57 x
IO'
7.66
x I
OJ
1.39
x IO
4 1.
85 x
lo4
2.21
x I
O4
2.49
x I
O4
2.70
x 1
0.
2.85
x I
O4
3.23
x 1
0.
3.34
x I
O4
3.66
x IO
* 3.
93 x
104
3.
93 *
101
cut-o
ff En
W8Y
. 0.
0478
0.
0478
0.
0141
0.
0485
0.
0443
0.
0217
0.
215
l&V
-
Tabl
e 11
.2. (r
n,lri
nlru
c/)
Twa0
-I BL
BJ
L BE
4.c
B&m
B&
c L
bb
, bb
, bh
m
&.a
3 ii so
urce
(S)
Fast
Muc
us
Slow
hlu
cua
Boun
d Se
ques
tere
d Al
Fa
st M
ucua
Sl
ow M
ucua
Bo
und
Sequ
osto
rad
Al
B -i h
0.01
0 0
0.01
5 0
0.02
0 0
0.02
5 0
0.03
0 3.
56 x
lo1
0.03
5 9.
12 x
IO
4
0.04
0 1.
22 x
10
0.
045
4.00
x I
O4
0.05
0 8.
50 x
IO
*
0.05
5 1.
42 x
10-
J
0.06
0 2.
06 x
10
0.06
5 2.
64 x
lo-
0.07
0 3.
09 x
10
0.07
5 3.
39 x
lo
0.
080
3.56
x 1
0-l
0.08
5 3.
61 x
lo-
0 5.
00 x
lo
0
4.99
x l
o
0 4.
97 x
10
6.43
x 1
0.
4.96
x I
O
4.13
x I
O3
4.96
x I
O
2.65
x I
O
4.95
x l
o
6.50
x l
Oa
4.92
x I
O-
1.08
x I
O-
4.85
x lo
I.55
x lo
- 4.
73 x
lo-
1.99
x l
o-
4.56
x l
o
2.45
x l
o
4.35
x l
o
2.88
x J
O-
4.14
x I
O-
3.22
x 1
0-l
3.94
x l
o
3.46
x I
O-
3.77
x l
o
3.59
x l
o
3.63
x l
o
3.64
x l
o
3.51
x l
o
0 0 0 0 0 4.
40 x
10.
7.
96 x
10.
7.
34 x
10
2.
39 x
IO
= 4.
98 x
1O
J 7.
88 x
lo*
1.
09 x
IO
- 1.
34 x
lo-
1.
51 x
lo
1.
61 x
IO
1.
65 x
IO
-
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0
3.98
x l
o
0
0 2.
08 x
10.
3.
97 x
lo-
0
3.40
x l
o4
9.72
x l
Oa
3.95
x I
O-
0
3.45
x 1
0-J
5.60
x l
Oa
3.88
x l
o
1.40
x l
o=
4.30
x I
OJ
1.22
x I
O
3.66
x 1
0
1.90
x 1
04
1.28
x l
o
1.79
x l
o
3.30
x 1
0
5.82
x 1
OJ
2.06
x l
o
2.22
x l
o
2.97
x I
O
9.69
x l
Oa
2.45
x I
O
2.49
x 1
0-l
2.70
x I
O
1.17
x I
O
2.54
x I
O
2.55
x l
o
2.53
x I
O
1.22
x I
O
2.44
x l
o
2.49
x I
O-
2.39
x I
O-
1.20
x I
O
2.33
x l
o
2.35
x I
O-
2.27
x I
O
1.22
x l
o
2.18
x l
o-
2.20
x l
o-
2.15
x I
O-
1.24
x l
o
2.02
x l
o
2.04
x l
o-
2.05
x 1
0-l
1.25
x l
o
1.87
x I
O
1.88
x I
O
1.92
x l
o-
1.25
x I
O-
1.74
x l
o
1.75
x I
O
1.80
x l
o
1.21
x l
o
1.63
x l
o
1.63
x l
o-
1.69
x l
o
1.18
x l
o
0 0
0 5
0 0
0
0 n
0 A
0 P
1.73
x 1
0
7 1.
84 x
IO
$
5.25
x I
O
,J
1.02
x I
O4
1.62
x l
o4
2.28
x l
o+
2.98
x l
O*
3.71
x l
o4
4.44
x 1
04
5.17
x I
O4
-
ABSORBED FRACTIONS FOR ALPHA. ELECTRON AND BETA EMISSIONS 469
7 o~~~bbbbbbbb%gbb%%bb~ e-------- w---e------- x x x x x x x x * x x x x * x x x x x x x
xxx*xxxxx*xxxxxxxxxxx
-
470 THE REPORT OF A TASK GROUP OF COMMIITEE 2
(H25) The energy spectra of each of the beta particle emissions listed in Table H.3 for negatrons, and Table H.4 for positrons, were obtained from the National Nuclear Data Center (Brookhaven National Laboratory). Each spectrum was calculated from the Evaluated Nuclear Structure Data Files (ENSDF), using the computer code FUDLST (Burrows, 1988). In order to evaluate the above integral, at least 150 energy bins were used to represent each energy spectrum.
(H26) Table H.5 gives the spectral-average absorbed fractions, as a function of mean spectral energy, for each of the negatron emissions listed in Table H.3.
(H27) Table H.6 gives the absorbed fractions calculated in the same manner for the positron emissions listed in Table H.4. For positron emissions of energy higher than the maximum listed value of 0.7353 MeV (from IsO), the absorbed fractions are identical to the values calculated for negatrons.
Table H.3. Beta-emitting isotopes with their average emitted energy used to evaluate
AF(T- S)d- for jJ - emissions
lsotopc numc Avcragc energy (McV)
H 16R Ni WRu 5s C Pm 4sCil T-C Co 12Sn *Sn *Sn Yr IMTI Cl Kr ?lllBi 24Na Sr yY 32p P3c *Y Al Kr AS
0.0056 0.0 IO0 0.0174 0.03 Is* O.OJH7 O.OlJX 0.06 I7 0.0773 0.0x45 o.ousn 0.1354 0.1581 0.1604" 0.2010 0.2394 0.2507 0.2562 0.38X8 0.55 16 0.5672 0.5886 0.691H 0.x153 0.9258 1.2350 1.6938 1.8570
Branch-energy selected from complex spectrum.
-
ABSORBED FRACTIONS FOR ALPHA, ELECTRON AND BETA EMISSIONS
Table H.4. Beta-emitting isotopes with their average emitted energy used to evaluate
AF(T- S)p+ for B l emissions
471
Isotope name Average energy ( MeV)
JYCl Cr 3b 7Cd JC0 L:Na -2Mn F r*Y JN 0
0.0502 0.09 I4a 0.1300 0.1418 0.2013 0.2 I55 0.24 16 0.2498 0.3595 0.4918 0.7353
Branch energy selected from complex spectrum.
H.6. Algebraic Approximations
(H28) To facilitate the evaluation of AF(T- S), for emissions of intermediate energy, and for complex emission spectra involving multiple radiation energies, the calculated values of AF(T+S), are conveniently represented as functions of radiation energy by fitted algebraic expressions. Appriopriate expressions, which were developed by A. Birchall and N. S. Jarvis (National Radiological Protection Board), are given in Table H.7. Values of the parameters to be substituted in thcsc expressions, in order to evaluate absorbed fractions for emissions of alpha particles, electrons, negatrons, and positrons, respectively, are listed in Tables H.8-H. II. The approximate values so obtained represent the calculated values of absorbed fraction to within about f 1% relative error. As noted above, however, other approximations to the calculated values of AF(T-S), may also be appropriate.
References
Bielajew. A. F. and Rogers, 0. W. 0. ( 1991). PRESTA. The paramctcr rcduccd electron-step size transport algorithm for electron Monte Carlo transport. National Research Council of Canada. Publication PIRS- 0042.
Burrows. T. W. (1988). The program RADLST. Brookhaven National Laboratory. New York. information Analysis Center Report. BNI-NCS-52142.
ICRU ( 1984). Slopping bwers jar &lecfrons and Positrotts, International Commission on Radiological Units and Measurements Report 37. ICRU Press. Bethesda, MD.
ICRU (1993). Stopping Pours and Runges fir Prorons and Alphu )Irrricles. International Commission on Radiological Units and Measurements Report 49. KRU Press, Bethesda. MD.
James. A. C.. Gehr. P.. Masse, R.. Cuddihy. R. G.. Cross, F. T.. Birchall. A,, Durham. J. S. and Briant, J. K. (1991). Dosimetry model for bronchial and extrrthoracic tissues of the respiratory tract. Rudiur. Prot. Dosim. 37.22 I-230.
Nelson. W. R.. Hirayama. H. and Rogers, 0. W. 0. (1985). The EGS4 code system. SLAC-Report-265, Stanford Linear Accelerator Center.
-
Tabl
e H.
5.
Valu
es o
f abs
orbe
d fra
ctio
n. A
F(T-
S)J-
. fo
r ne
gatro
ns
6
Sour
ce(S
) ET
, ET
2 ET
2 ET
2 B&
I BB
b,
B&VU
B&
a B&
s 4 8
Targ
et (T
) SU
lf&X
Surfa
ce
Boun
d Se
pues
tero
d Fa
rt Sl
ow
Boun
d Se
ques
tere
d AI
M
llc~
MU
CU
S i n
Aver
age
Ener
gy,
MC
V
0.00
56
0 0
1.82
x lo
- 2.
81 x
lo-
0.01
00
0 0
1.81
x lo
- 5.
91 x
10-J
0.
0174
1.
17 x
104
1.94
x lo
1.
71 x
lo-
4.87
x lo
0.
03 I5
1.
76 x
IO
6.60
x lo
- 1.
48 x
lo-
9.07
x lo
0.
0487
4.
70 x
lo-
2.48
x lo
- 1.
20 x
lo-
9.41
x lo
0.
0498
4.
56 x
lo-*
2.
32 x
lo-
1.22
x lo
9.
71 x
lo
0.06
17
5.60
x lo
-2 3
.43
x lo
-2
1.04
x lo
- 8.
44 x
105
0.07
73
6.00
x W
2 3.
90 x
lo
9.23
x lo
- 7.
78 x
lo-
0.08
49
5.86
x 10
-2 3
.91
x lo
-2
8.62
x IO
7.
27 x
lo
0 0 4.
42 x
10
1.
62 x
lo
5.33
x 10
-Z
5.08
x W
2 6.
82 x
lo
7.54
x lo
7.
47 x
10
0 0 1.
97 x
IO-
1.91
x lo
- 5.
64 x
lo-
5.42
x 10
7.
06 x
lo-
7.74
x 10
7.
64 x
10-2
2.5
0 x
lo
2.50
x lo
- 2.
46 x
lo
2.21
x lo
- 1.
86 x
10-l
1.89
x lo
- 1.
63 x.
lo
1.46
x lo
1.
36 x
10-l
1.01
x 10
4 4.
96 x
lo-
2.24
x lo
- 7.
49 x
lo-
9.67
x lo
- 9.
85 x
lo-
9.53
x Ia
2 9.
34 x
lo-
8.90
x lo
-
0 h)
0 0 0 0 0 0
-
ABSORBED FRACTIONS FOR ALPHA, ELECTRON AND BETA EMlSSlONS 473
0
4 0
X
% od
X
X
x
X
-
474 THE REPORT OF A TASK GROUP OF COMMIITEE 2
-
0.09
58
2.15
x l
o
2.22
x lo
- 2.
69 x
lo-
1.32
x lo
0
9.49
x lo
- 9.
69 x
lo-
1.
15 x
lo-
6.
87 x
lo-
9.72
x l
Oa
0.13
54
1.77
x I
O-
1.80
x lo
- 2.
07 x
lo-
1.19
x lo
1.
24 x
lo-
6.76
x 1
8
6.85
x lo
-* 7
.95
x lo
-*
5.12
x lo
- 1.
21 x
lo
0.15
81
1.57
x lo
- 1.
60 x
lo-
1.80
x lo
1.
10 x
lo
2.
86 x
lo-
5.66
x lo
- 5.
72 x
lo-2
6.5
8 X
lo
4.37
x l
O-*
1.3
0 x
lo-
0.16
94
1.49
x lo
1.
51 x
lo
1.69
x lo
- 1.
05 x
10
3.
88 x
IO
- 5.
21 x
lo-
5.26
x l
Wz
6.04
x 1
0
4.07
x lo
- 1.
34 x
lo-
0.
2010
1.
30 x
lo
1.32
x I
O-
1.44
x 1
0-l 9
.60
x lo
- 5.
83 x
lo-
4.21
x lo
- 4.
24 x
lo
4.81
x 1
04 3
.38
x lo
1.
43 x
IO
- 0.
2394
1.
08 x
lo-
1.09
x lo
- 1.
19 x
lo-
8.10
x lo
- 1.
26 x
lO
A 3.
39 x
lo
3.41
x lo
-*
3.87
x l
o
2.74
x 1
0-*
1.49
x l
o-
0.25
07
1.08
x l
o-
1.08
x 1
0-l
1.17
x lo
- 8.
18 x
lo
1.
17 x
lO
a 3.
25 x
lo-*
3.2
6 x
lo-2
3.6
6 x
10-2
2.6
7 x
lo-
1.50
x l
o
0.25
62
1.03
x lo
- 1.
04 x
lo-
1.12
x lo
7.
87 x
lo-
1.27
x 1
0d 3
.11
x lo
- 3.
12 x
lo
3.
53 x
lo-2
2.5
5 x
l@*
1.52
x lo
- 0.
3888
6.
62 x
lo
6.66
x lo
- 7.
15 x
lo-
5.28
x lo
- 2.
89 x
lO
A 1.
86 x
lo-*
1.
87 x
10-
2 2.
09 x
10-
2 1.
57 x
lo
1.
62 x
lo-
0.
5516
4.
60 x
lo
4.69
x lo
4.
88 x
lO
- 3.
83 x
1~
3.93
x
10d
1.19
x IO
-* 1
.20
x IO
-2 1
.32
x lo
-2 1
.06
x 10
-2 1
.68
x 1~
0.
5672
4.
39 x
lo
* 4.
51 x
lo
4.69
x lo
3.
66 x
10
4.
11 x
10-
1.
16 x
lo-
1.17
x 1
0-2
1.30
x 1
0-2
1.02
x l
@*
1.68
x 1
0.
0.58
86
4.20
x l
o-
4.34
x lo
4.
49 x
lo-*
3.5
1 x
lo-
4.24
x 1
0d 1
.12
x lo
- 1.
12 x
lo-
1.
24 x
l@
* 9.
84 x
l@
1.
69 x
10.
0.
6918
3.
50 x
lo-
3.67
x 1
05 3
.73
x lo
2.
98 x
IO
- 4.
69 x
10d
9.2
0 x
lo-
9.25
x 1
0-l
1.02
x lo
-*
8.26
x 1
0 1.
71 x
lo-
0.
8153
2.
87 x
lo-
3.04
x lo
- 3.
07 x
lo-
2.47
x l
o-
5.17
x l
OA
7.73
x I
O-
7.79
x l
@
8.51
x 1
0-3
7.02
x lo
1.
72 x
lo
0.
9258
2.
47 x
lo-
2.61
x lo
- 2.
64 x
lo
2.14
x 1
0
5.50
x l
OA
6.84
x lo
- 6.
91 x
lo
7.
53 x
10
6.
25 x
10-
3 1.
73 x
10-
J 1.
2350
1.
76 x
lo-
1.86
x l
o-
1.89
x lo
-*
1.55
x lo
6.
04 x
lo-
5.24
x lo
5.
29 x
10-
5.
71 x
lo-
4.89
x 1
0-
1.75
x 1
0
1.69
38
1.22
x lo
-*
1.28
x lo
- 1.
32 x
lo-
1.10
x lo
- 6.
55 x
10d
4.1
4 x
lo
4.19
x I
@
4.48
x I
O-
3.93
x l
@
1.76
x l
o
1.85
70
1.11
x lo
1.
16 x
lo-
1.19
x lo
- 9.
97 x
10
6.
69 x
10d
3.9
0 x
lo-
3.95
x 1
0
4.21
x 1
0-3
3.72
x 1
0-3
1.76
x l
o
-
Tabl
e H.
6. V
alue
s of
abs
orbe
d fra
ctio
n,
AF(T
- S)
b..
for
posi
trons
ET?
ml
BBb,
BB
b,
BBb,
Bs
, Bh
sour
ce
(S)
sulfe
ee
surfa
ce
Boun
d Se
ques
tertd
FM
t
hluc
ua
Slow
M
UCUS
Boun
d Se
quea
tmd
Al
o.os
o2
3.23
x I
O*
1.26
x l
o*
1.32
x l
o
0.09
14
6.77
x I
i* 4.
30
x 10
-J
9.07
x l
o+
0.13
00
6.53
x I
O*
4.59
x
loa
7.14
x
10s
0.14
18
6.20
x 1
0
4.44
x
IO5
6.67
x 1
0J
0.20
13
4.49
x 1
OJ
3.24
x
10
4.83
x
IO*
0.21
55
3.94
x I
OJ
2.82
x
IO=
4.32
x
10J
0.24
16
3.74
x I
OJ
2.69
x
10J
4.02
x
IO*
0.24
98
3.48
x l
o*
2.48
x
IO5
3.81
x
lo=
0.35
95
2.46
x 1
0J
1.74
x 1
0
2.68
x
lo*
0.49
18
1.68
x 1
04
1.18
x 1
0
1.90
x l
o*
0.73
53
1.06
x l
o4
7.S7
x l
o=
1.24
x I
O*
1.11
x l
o
8.25
x l
o*
6.48
x
IO3
6.04
x
10J
4.39
x 1
OJ
3.94
x 1
03
3.68
x
10
3.49
x 1
04
2.49
x 1
0J
1.78
x I
O*
1.18
x 1
0J
3.12
x
lo*
3.55
x
1Oj
2.03
x
lo-
1.03
x l
o-
8.42
x 1
0s
8.65
x 1
04
1.46
x l
o-
1.02
x I
O-
8.57
x
lo4
8.69
x 1
0J
1.14
x l
o-
8.77
x
lo4
8.22
x 1
0
8.32
x 1
0J
1.06
x l
o
8.27
x 1
0
6.08
x I
O*
6.13
x 1
Oj
7.41
x
104
6.08
x
lOa
5.35
x 1
0J
5.40
x 1
0J
6.58
x
103
5.41
x I
O*
5.11
x 1
0J
5.15
x 1
04
6.08
x
10-z
5.
11 x
10-
2
4.75
x
10
4.79
x
10
5.75
x I
O*
4.80
x
10
3.42
x
10
3.44
x
lo5
3.91
x
lo4
3.42
x
10
2.35
x 1
0J
2.36
x
lo5
2.70
x
IO*
2.37
x
lo4
1.47
x l
o*
1.49
x 1
0
1.69
x l
o-
1.51
x I
O5
3.44
x
IO-
1.87
x l
o
1.24
x l
o
4.30
x
lo
1.49
x 1
0
3.08
x
lo
3.45
x
lo
4.72
x
10
9.76
x
10
1.73
* l
o4
2.43
x
lo4
-
B&e
BL
BBuo
bb
, bb
, bb
, ha
sour
ce (S
) FU
t M
ucus
Sl
ow
Muc
us
Boun
d Sq
uest
ercd
Al
Fu
t Sl
ow
Boun
d Se
ques
tere
d Al
M
ucw
Muc
us
o.os
o2
1.91
x lo
- 2.
18 x
lo
4.26
x l
o-
0.09
14
2.63
K lo
2.
71 x
lo-
3.16
x lo
- 0.
1300
2.
31 x
lo
2.
34 x
lo-
2.47
x lo
- 0.
1418
2.
16 x
lo
2.18
x 1
0-l
2.28
x lo
0.
2013
1.
50 x
lo
1.51
x 1
0-l
1.58
x lo
- 0.
2155
1.
31 x
lo-
1.32
x I
O-
1.40
x 1
0-l
0.24
16
1.23
x lo
1.
23 x
lo-
1.29
x l
o-
0.24
98
1.14
x lo
1.
15 x
lo-
1.21
x lo
- 0.
3595
7.
85 x
10
7.
85 x
lO
a 8.
19 x
10
0.
4918
5.
31 x
1O
J 5.
35 x
1O
J 5.
60 x
lo*
0.73
53
3.28
x 1
0
3.45
x l
Oa
3.48
x 1
0
8.16
x 1
OJ
6.38
x lo
1.
45 x
lo
3.38
x 1
0-L
1.
47 x
lo
1.83
x lo
- 1.
42 x
lO+
6.96
x lo
1.
09 x
IO
- 2.
87 x
lo
9.72
x 1
0
6.05
x lo
9.
33 x
104
6.
78 x
10
8.
71 x
lOa
9.33
x l
o-
6.38
x l
Oa
1.94
x lo
4 4.
40 x
10
3.
43 x
104
2.
81 x
lo*
4.83
x 1
0.
1.80
x I
O-
1.91
x l
o
2.39
x lo
- 1.
25 x
IO
- 1.
27 x
lo
1.39
x I
O
8.64
x I
O=
8.65
x 1
Oa
9.25
x 1
Oa
7.69
x I
O*
7.68
x 1
0
8.23
x 1
0
4.69
x l
Oa
4.68
x l
Oa
J.08
x I
O-
4.05
x 1
0
4.0s
x t
o=
4.44
x I
Oj
3.S9
x l
Oa
3.S8
x lo
* 3.
90 x
IO
* 3.
39 x
1O
J 3.
39 x
lO
a 3.
72 x
IO
2.
02 x
IO
4 2.
02 x
10
2.
21 x
104
1.
36 x
lOa
1.37
x lO
a 1.
50 x
10
8.
50 x
lO
a 8.
55 x
IO
= 9.
33 x
lo=
1.03
x lo
2.
28 x
lo4
8.66
x 1
0J
7.72
x I
O4
6.70
x l
o=
1.09
x 1
0
6.06
x 1
OJ
1.16
x 1
0
3.83
x l
Oa
1.40
x 1
0d
3.31
x 1
04
1.4s
x I
O
3.00
x I
O=
1.48
x 1
0
2.81
x l
Oa
I.50
x 10
4 1.
76 x
IO
= 1.
61 x
10
1.
20 x
10J
1.
66 x
10
7.
71 x
IO
J 1.
71 x
10
-
Tabl
e H.
7. A
lgeb
raic
fu
nctio
ns u
sed
to a
ppro
xim
ate
AF(T
-S)
for
alph
a-,
elec
tron-
, ne
gatro
n- a
nd p
ositr
on-
emitt
ing
sour
ces
Func
tion
2
AF(r)
-
[ 1 -
e-*
~a]
[a,(l
+
a#
+ a,
e*
+ a.
e
+ a,
c)]
Func
tion
3
-
Tabl
e H.
X.
Fitte
d va
lues
of
par
amet
ers
fur
suhs
tirui
ion
in a
lgrh
raic
fu
nctio
ns
to a
ppro
xim
ate
AF(T
- S)
,, fo
r al
pha-
rmill
ing
stn~
rces
CLFC
FU
UCtio
ll Fi
ned
V~~I
C of
Fol
lowi
ng
Para
met
er
I 2
3 4
5 6
7 8
9 IO
II
12
I I 3 I I I 3 I I I 3 I I I 3 I I
0.01
7%14
5 -3
119.
876
0.19
0049
9 0.
08llS
nS
-332
.281
-0
.043
7065
6 0.
0385
861
-177
3.79
7 0.
1467
869
0.05
5917
18
0.00
1987
374
-560
1.58
5 0.
1983
734
0.06
4996
44
-917
.118
6 -0
.033
lOlS
7 0.
0217
9814
-2
029.
563
0.14
0741
0.
027S
7517
0.18
18
0.07
1772
34
2.16
186
0.41
4282
3 0.
67Sl
693
0.9U
8594
0.
0116
813
7.59
OlS
2 0.
1566
737
-
0.00
4805
373
-432
.934
8 0.
2460
405
0.06
2665
83
-48.
0913
3 0.
2673
197
0.05
4824
65
-26.
6463
4 0.
2542
903
0.03
3302
28
.884
.288
0.
0816
4135
-179
2.14
9 0.
0764
8437
-304
.843
6 0.
3973
314
0.06
9833
29
-118
4.59
3 0.
0874
2065
0.
0240
528
-252
6.06
9
0.06
9921
42
-IO36
153
0.09
1662
63
0.03
1967
2 -2
478.
645
0.25
0.
0156
3289
6.
C352
4l
0.36
5012
7 l.s
3643
1
0.02
2l66
91
-394
.886
9 O
.ll27
097
O.l3
5027
3 -1
69.1
389
0.14
7914
9 0.
006S
3401
4
0.17
a478
9 0.
0076
9929
0.27
0432
6 0.
0352
6641
0.26
5075
5 0.
0072
6001
I
-121
65.9
2 0.
2029
002
0.11
4287
2
4477
.944
0.
2313
976
0.11
3453
8
8.84
8307
0.
08l7
0446
-
-470
.537
5 0.
4806
218
0.07
4649
73
416.
8051
-0
.032
1554
9
-376
.569
3 -0
.035
56l5
8
-133
.05u
-0
.052
5050
2
.
0.17
5204
3 -3
81.9
491
0.15
8163
0.
0158
9757
-1
162.
612
0.39
1113
4 0.
2871
758
.tii.n
57
0.09
2728
64
-311
.263
8 0.
1371
357
-0.0
1108
624
-182
4.43
2 0.
1549
19s
0.20
9227
7 -2
9.22
377
0.5
0.26
6322
2 1.
2479
13
0.39
5056
2.
5955
32
0.23
4951
7 0.
0937
5M3
6.07
3588
0.09
8114
%
-390
.233
3 0.
1676
271
0.01
9688
22
-859
.253
1 0.
2946
132
0.13
2886
4 -1
63.7
452
-0.0
1612
15
0.08
1755
%
0.00
9140
343
0.02
5271
41
0.08
536O
a3
- -8
.931
576
0.03
5131
11
= IO
-S28
.738
8
-353
.292
1
-322
1.80
7
0.2n
osl6
0.38
749'
2
-0.0
5428
12
0.04
3OSl
29
-394
.261
4 0.
4336
853
0.07
3268
54
-152
.643
0.
3898
331
O.l7
0419
8 -4
4.87
835
0.00
7152
097
-1U9
.924
0.
3744
433
0.18
9707
5 -3
0.60
365
0.32
9908
0.
0506
0541
-1
13.7
795
0.21
0405
7 0.
0051
3349
7
0.31
1957
8 -4
.889
305
I lo
0.24
0524
-
0.18
3926
7 O
.OZl
l418
2
0.06
6623
89
l.811
3$7
-741
.428
2 0.
2740
834
-580
.599
5 0.
1890
273
0.4
0.23
3037
5 1.
6069
37
0.69
4407
6 I.6
1783
5 0.
2810
387
-0.0
2880
487
7.88
7438
0.06
0820
I5
-217
.541
3 0.
4092
593
0.09
193%
27
-25.
7198
7 0.
1000
563
0.01
1882
4 -1
568.
605
9.51
3849
-7
92.4
591
0.21
9184
8 9.
6216
04
-133
.714
5 -0
.I589
l39
3.21
2547
-3
09.0
881
-417
.279
5 0.
1039
428
-149
7.62
0.
3003
989
x lo
" x1
0-
x IO
' x
IW'
-
T&k
H.9.
Fi
tcal
va
lurs
of
par
amcr
rrs
for
suhh
turh
in
slp
zhra
ic
func
tions
IO
app
roxi
mat
e AF
(T-
SI,
for
elrc
lron-
cmitt
iug
sour
ces
Cl&C
Fu
nctio
n Fi
tted
Valu
e of
Fol
lowi
ng
Para
met
er
ET, -
.mfa
ce
ET+a
face
ET+a
md
ET*-
seq.
B&-fa
st
BB,,&
ow
B&-b
ound
Bb-s
eq.
B&-A
I
BB,-f
pst
BB,,-
slow
BB,,-
boun
d
BB,-s
eq.
BB,,-
AI
bb,-f
pst
bb,-s
low
bb&m
md
bb,-s
eq.
bb,c
AI
1 1 3 I I I 3 I 2 1 I 3 I 2 1 1 3 1 2
0.06
5053
-3
54.4
06
2.01
4387
0.
0363
35
-991
.429
2.
1254
58
O.O
lY84
2 -2
4.63
74
1.58
0302
0.
0460
65
0.00
9434
-1
8.25
63
1.52
1565
0.
0109
17
-I 13
4.08
2.
1080
72
0.03
0823
-8
0.07
44
I.788
97
0.03
9695
0.18
18
0.01
2801
0.
7908
06
13.4
5734
1.
1627
24
53.7
0593
-0
.009
02
11.5
1802
10
.273
88
-
0.03
6493
-1
7.09
9 1.
9570
65
0.01
2268
-4
90.9
83
2.37
2073
0.
0139
01
-80.
9776
1.
7667
27
0.12
1082
0.02
2208
-2
3.54
3 1.
5268
98
0.05
5161
-6
78.6
04
2.07
6011
0.
0520
48
-89.
4917
1.
7723
34
0.07
7509
0.02
4444
-2
3.37
06
1.56
2825
0.
0287
98
-705
.929
2.
1233
17
0.05
7382
-8
7.78
9 1.
8082
63
0.08
3806
0.25
0.
0233
37
0.86
7548
14
.665
98
1.38
1178
33
.946
87
-0.0
168
8.05
6959
11
.691
28
-
0.08
4383
-4
3.11
91
1.91
8628
0.
1179
06
-169
.24
2.23
4064
0.
0014
11
-86.
00 1
1.
2630
35
0.00
9699
0.00
0393
-4
00.4
56
-1.8
0563
77
7.72
34
-1.7
32
3.73
3329
-1
0.24
82
-379
.908
-1
.662
48
0.28
7203
0.21
9072
-1
80.3
89
2.12
3581
0.
0722
01
-21.
9379
1.
7244
34
0.01
0307
-1
090.
27
2.32
3796
0.
1657
66
0.19
0302
-1
31.2
82
2.11
3636
0.
0612
25
-18.
8215
1.
7920
23
0.04
9712
-3
94.0
19
2.35
0691
0.
1579
4
0.5
0.07
042
0.88
6749
19
.058
02
1.54
244
27.3
7533
-0
.032
74
6.96
8345
13
.335
13
-
-0.2
0809
-5
72.7
2 2.
1745
99
0.27
8314
-4
76.0
06
2.16
8483
0.
1252
74
-69.
2237
1.
9184
43
0.05
8332
0.00
0786
-8
59.7
4 -1
.614
89
1689
.772
-1
.573
64
124.
4556
-2
5.11
86
-832
.213
-1
.533
68
0.28
7178
0.11
2968
-3
58.2
16
2.39
3708
0.
1316
71
-170
.786
2.
2684
27
0.01
5445
-1
8.79
36
1.87
0242
0.
1019
48
-60.
4808
2.
0886
22
0.02
5629
-5
38.8
17
2.53
5345
0.
1757
29
-48.
4179
2.
2578
59
0.01
8913
-1
6.76
86
2.05
5972
0.
0717
51
-189
.165
2.
3443
06
0.4
0.04
8063
0.
7114
81
149.
3354
1.
6212
45
45.5
38
-0.0
3676
6.
9606
92
22.5
9052
-
I
0.05
9129
-1
39.7
56
2.15
3796
0.
0596
99
-53.
6711
2.
0263
1
0.01
0356
-1
6.87
11
1.82
904
0.08
5985
-3
06.9
2 2,
3699
79
0.00
1772
-2
.644
63
-16.
7712
1.
1166
24
-2.1
1559
3.
2915
14
-46.
8517
-1
.275
88
-2.1
1622
0.
04
-
- 100
.628
I .
8355
25
-234
.006
I.9
5968
4
-89.
8524
2.
2864
18
-288
.868
1.
9573
59
-281
.914
1.
9972
5
-16.
5773
1.
4143
84
-67.
5677
I.9
39
-46.
6822
1.
9638
19
-21.
8086
I .
6898
66
-
t&e
Func
tion
I 2
3
Fitte
d V&
e of
Fo
llowi
ng
Para
met
er
4 5
6 7
8 9
IO
II I2
BB,&
uc
B&&W
BB,&
ound
BI&-
wq.
B&-A
l
BB&f
ut
B&&W
BB,-b
ound
BfL-
lcg.
BB
,&At
bb,-f
-1
bb,-s
low
bb,-b
ound
bb,-r
eq.
I I 3 I I I 3 I 2 I I 3 I 2 I I 3 I
0.00
7260
547
-214
.975
2.
1386
41
0.01
3313
65
-15.
60-1
2 1.
8455
84
0.18
18
o.oO
5491
0.
6245
98
0.01
80 I4
72
-17.
2882
2.
1750
34
0.04
2368
o.oo
1949
13.8
6767
-4.8
8363
4 x
lOA
-72
4619
2.
2777
61
0 02
2782
-2
2.09
16
I .69
1069
-197
.04
2.44
O31
5 0.
0278
93
-32.
53 1
9 2.
0808
03
0.91
1726
I0
9 23
02
0.00
6082
a
a507
2 26
.444
96
-299
2.34
1 I.4
4166
3 0.
0 I3
695
-17
2668
2.
1759
7
0.01
0659
45
-21.
24Q
5 1.
5ol1
57
0.00
7667
-I4
4 42
2 2
4023
5
0.00
1019
174
-461
.785
t .
6a96
8 o.
oo33
a9
-237
.087
2.
5202
64
0.25
0.
0522
29
I.052
155
Il.14
68
I.111
43
63.9
6697
0.03
7944
-1
7.76
81
I.987
425
0.07
5468
-2
8.39
41
2 37
9867
-31.
5976
2.
0643
26
-3 I
.044
2 2.
0973
91
I.387
472
5.61
7411
-0. I
7368
0.
4102
69
o.oo
o393
-3
99.0
76
-1.9
2481
77
6.02
08
-I 86
139
1074
593
0.07
3701
0.07
5876
-0.0
2104
3.44
5695
* IO
4
-13.
1079
-3
78.6
07
-I .7
9829
0.
1254
93
0.14
3302
-1
7.75
31
2.19
0977
0
0068
98
-27.
6829
18
1579
7 0
0348
8 I
-32
2445
1.
8645
87
O.lo
6393
-5
0.98
56
2.05
9853
0.
0875
23
-20.
5774
I
7945
22
0 17
8955
-7
9.91
2 2.
3615
94
0.5
0.35
9364
0
9805
82
29.6
o645
1.
7264
7 J8
148
44
-0.0
6287
1.
4409
5.
4342
51
0.00
1561
-4
22.1
27
I .6a
267
0.12
8516
-2
6. I2
7 2
1054
97
0.01
5492
-1
92.2
27
2.57
3017
O.o
oO78
6 -8
58.9
87
-I .8
0744
16
89.3
34
-1.7
6746
I.4
9134
1 -1
7.02
19
-831
.919
-I
.727
68
0.11
8654
-4
2.98
86
2.54
2548
0.
0171
49
-163
.798
2.
3645
03
0.00
2686
-0
.074
16
I .99
9984
0.00
9441
-l0
7.09
8 I 9
677o
a 0.
0279
29
-I 17
.991
2.
3467
19
0. I4
8723
-2
0.62
29
2.49
0604
0.4
0.38
4747
1.
0810
31
117.
5969
0.
2780
79
I II
16.3
4 0.
01 I7
72
3.93
2391
25
.072
08
0.00
7053
.5
17.2
97
2.36
0246
0.
0527
97
-27.
4988
2
1313
17
0.00
355
-4.0
4509
1.
5507
54
0.02
2747
-5
9.75
927
I941
653
o.oO
3504
-1
961.
651
2.16
8247
0.05
804
-42.
8039
8 2.
5875
52
o.oo
2144
-1
1.89
227
I.485
352
0.01
4213
-1
8.83
598
I .53
6484
0.00
5617
-3
0.82
74
I .70
29%
0.07
7187
-J
7.93
59
2.25
0686
0.
0325
24
-IJI.S
J2
2.68
8903
0.02
262
0.13
5082
-17.
lIY3
1.58
2392
0.06
4615
-2
7.53
46
2.11
5816
0.00
1544
-3
.062
I8
0.26
6570
6
0.06
1775
-4
1 82
98
2 52
7532
bb,-A
l 2
o.oo
1772
I.1
8713
2 -6
.502
16
-0 9
3579
-4
.923
43
-1.5
S613
-1
2 30
08
-0.0
4425
-I
1170
9 0.
0282
17
-
Tabl
e H.
I I.
Fitte
d va
lues
of
para
met
ers
for
subs
titut
ion
in a
lgeb
raic
fu
nctio
ns
to a
ppro
xiam
te
AF(T
- S)
p. f
or p
ositr
on-e
mitt
ing
sour
ces
caw
FUlK
li0n
Fiid
V&
e of
Fol
lowi
ng
Panm
etcr
I 2
3 4
5 6
7 8
9 10
B&-fa
st
B&-S
bW
BB&O
WJ
WDC
WP.
B&
-Al
bb,-f
asl
bb,-*
loa
bb,-b
arad
kc-u
q.
I 0.
0532
2816
t 0.
0213
I926
3 0.
1818
1 0.
1125
952
-21.
7145
1 2.
0019
1 0.
0171
4905
-1
30.3
817
1.96
4898
-42.
1711
6 1.
9732
19
0006
3643
13
-18.
8606
1.
38S2
8 0.
0391
2S37
0.
7279
522
62.O
7332
u3
on
?0.0
%61
-I8
.7l8
9l
2.4o
lo96
oa
oi64
67m
-4
95.9
539
1.8.
6378
5
t od
8on5
2 -3
0.55
993
2.32
im
0.00
337l
729
-12.
2U36
1.
0311
05
1 0.
0315
1113
6 -9
3.69
928
1.95
5103
0.
0029
?028
-2
43.6
406
I.405
51
3 0.
25
0.04
072J
82
0.74
684o
8 45
.666
27
1.21
0492
49
.992
81
I 0.
1082
752
-19.
9026
9 2.
1933
76
o.o0
5t93
tn
-284
.861
9 I.8
8931
2
2 o.
uoO
393
0.38
2015
2 .S
.U88
0?8
I5.6
4m3
-6.2
1192
3 -0
.506
4243
I 0.
2223
405
I 0.
2789
94
3 0.
5
I 0.
1456
224
2 O
.oO
O78
6
-21.
3l97
7 2.
1208
113
0.05
3wD9
5 -1
30.6
415
I.%?5
7 -3
5.08
35
2.12
3887
0.
0306
(369
-1
9.41
535
1.48
2641
0.41
6o?O
t 1.
2613
45
43.4
7673
0.
1699
746
39.8
7t67
-2
4.75
3 2.
0022
7 0.
0046
3291
6 -9
.655
8?3
1.18
2674
-8
63.2
721
-1.6
0625
6 16
97.4
92
-1.5
6977
? 1.
2501
45
I 0.
38on
52
I 0.
1912
379
3 0.
4
I 0.
1001
748
-30.
5599
3 2.
321m
o.
oO33
7172
9 -1
2.24
436
I.031
105
-3
4.05
969
2.33
0431
0.
OO
6o24
704
-8.2
rn I
I3
t.555
65J
0.36
7254
3 I.3
2143
2 71
.069
28
O.S
JO37
4 22
9.11
79
-30.
3826
6 2.
2631
86
0.00
518?
536
-5.4
6699
3 1.
8026
Sl
0.00
3121
981
-619
.581
1 I.8
5608
1
0.00
2588
461
9.48
9306
6.
7196
9
o.oO
2025
164
6.60
1281
1.
9913
95
0.05
9849
07
-19.
9143
4 I .
9?82
S7
0.00
5529
276
6.96
4317
1.
9489
o4
-0.6
8l77
57
-i6..1
%9?
-5
.144
O2
0.14
8889
2
0.01
6365
37
-383
.925
2 1.
8742
74
0.01
7337
94
a. 1
782
17
(I.51
3134
0.00
9628
979
-544
.225
1 I.8
6026
1
-I2.0
1085
-8
35.9
406
-1.5
3389
9
0.00
4494
271
5.85
6029
9.
#234
9
0.00
6954
227
-22
I .82
2 I
I .93
9597
0.14
6455
3
bb,-A
l 2
0.00
1772
2 1.
7692
39
-16.
12l4
7 8.
8769
24
-16.
7528
9 -1
1.95
439
-16.
1230
9 -0
.186
0699
-2
.244
41
0.03
3742
13