NBSIR 73-107 C^)

Improved Point Kernels for Electron

and Beta-Ray Dosimetry

Martin J. Berger

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

February 1973

Progress Report

Prepared for U. S. Atomic Energy Commission

Division of Biomedical and Environmental Research

Washington, D. C. 20545

NBSIR 73-107

IMPROVED POINT KERNELS FOR ELECTRON AND BETA-RAY DOSIMETRY

Martin J. Berger

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D. C.

May 1973

Progress Report

Prepared for U. S. Atomic Energy Commission

Division of Biomedical and Environmental Research

Washington, D. C. 20545

U. S. DEPARTMENT OF COWIIVIERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

IMPROVED POINT KERNELS FOR ELECTRON AND BETA-RAY DOSIMETRY

Martin J. Berger

National Bureau of Standards

Washington, D.C. 20234

A calculation has been made of the spatial distribution of absorbed

dose in a water medium around monoenergetic point-isotropic electron

sources. The calculation takes into account angular deflections and

energy-loss straggling due to multiple Coulomb scattering by atoms and

orbital electrons; it also includes the transport of energy by secondary

bremsstrahlung . The results are presented in the form of scaled point

kernels for 36 source energies between 10 MeV and 0.5 keV. The scaling

is done by expressing all distances in units of the electron mean range,

and makes possible easy interpolation to any source energy in the interval

covered. In order to illustrate the use of the point kernels, applications

are made to two problems arising in beta-ray dosimetry. The first problem

pertains to the self-absorption of energy in spherical source regions.

The other problem concerns the absorbe d- dose distribution as a function of

the distance from a semi-infinite uniform source region.

1. INTRODUCTION

This is a preliminary report dealing with the calculation of electron point

kernels in water. These kernels are functions which represent the distribution

of absorbed dose around point-isotropic sources of electrons. The purpose of the

report is to provide interim documentation for other papers on electron and betaray dosimetry which make use of these point kernels. Work on a detailed description of the calculation is in progress.

An extensive and systematic set of data on the distribution of absorbed dose

around point-isotropic electron sources is contained in the tabulation of

Spencer,—'^ whose results for a carbon medium have been used, with a slight scaling

adjustment, to produce point kernels for a water medium, both for monoenerge tic

21 sources and for a large number of radionuclide beta-ray sources.—

Spencer's calculations used the continuous- slowing-down approximation, i.e.

a schematization in which the rate of energy loss at each point along an electron

trajectory is assumed to be equal to the mean energy loss given by the Bethe

stopping power formula. Energy-loss straggling is disregarded in this approximation, which has the result that the amount of absorbed dose is slightly overestimated near the source and underestimated far from the source.

The calculations reported here are an improvement in the following respects:

(1) Energy-loss straggling is taken into account, bnth for energy losses

suffered in collisions with atomic electrons and for bremss trahlung losses.

(2) The effect of energy transport by secondary brems s trahlung on the noint

kernel functions is included.

(3) The calculations are extended to a low-energy region not previously

covered (25 keV to 0.5 keV)

.

2. SCALED POINT KERNEL

The distribution of absorbed dose in an unbounded medium, around a pointisotropic source emitting electrons of energy E , can be expressed in terms of

1
the specific absorbed fraction $(r,E^). This quantity is defined to be the

fraction of the emitted energy that is absorbed per unit mass of the medium at

a distance r from the point source. In order to minimize the dependence on

and thus to facilitate interpolation, it is convenient to express the specific

absorbed fraction in terms of a scaled point kernel F(r/r ,E ) defined by the

o o equation

2 F(r/r ,E )d(r/r ) = 4Trp$(r,E )r dr,

o O O O \ I

)

*/

where r is the c„s,d„a^— range at energy E^ and p is the density of the

medium. The scaling is accomplished by expressing distances in units of r^.

Table 1 gives the scaled point kernel F(r/r^,E^) for a water medium, for

36 values of the source energy E^ between 10 MeV and 0.5 keV. The results for

3 4/

E > 20 keV were obtained by a variant

— of the Monte Carlo method which

o combined random sampling with the use of the Be the stopping power theory as

formulated by Rohrlich and Carlson,—'^ the Landau—'^ energy-loss straggling distribution with the Blunck-Le isegang^''' binding correction, and the Goudsmitg / Saunderson— multiple-scattering angular distribution evaluated with use of the

9/ 10 11/ Mott— elastic scattering cross section and the Moliere ' screening cor- 12/ rection. For E <^ 20 keV another Monte Carlo model— was used in which all

o elastic collisions with atoms and hard inelastic collisions with atomic electrons

were followed by random sampling. The numerous soft inelastic collisions (giving

rise to atomic excitation or to the production of knock-on electrons with energies

smaller than 200 eV) were treated in the continuous-slowing-down approximation,

with the use of a "restricted" stopping power. The stopping power values at

energies below 10 keV were taken from a semi-empirical formula of Gree n"^"^

^ and

collaborators which gives results that are in reasonable agreement with experimental

* Continuous-slowing-down approximation

2
values of Cole ^^^ ^

Table 2 gives, for all 36 source energies, the values of the c.s.d.a. range

r and the values of the 90-percent ile distance x . The 90-Dercentile distance

o yO

is defined to be the radius of the sphere around a point source within which 90%

of the emitted energy is absorbed. Table 3 compares, for source energies of 1 MeV

and 0„1 MeV, the new scaled point kernel (with straggling effects) and the old

scaled point kernel (without straggling effects). Table 4 compares the 90-percentile distances obtained with and without inclusion of straggling effects.

3. ABSORPTION OF ENERGY IN A SPHERICAL SOURCE REGION

It may happen, either as the result of deliberate tagging procedures-—^—^'^

in radiobiological experiments, or as the result of accidental ingestion of

radioactive material, that radionuclides get incorporated into the genetic

material of the cell, particularly the cell nucleus. It is then of interest to

know what fraction of the emitted beta-ray and electron energy is absorbed within

the cell nucleus, and what fraction is absorbed in the surrounding - and presumably

less sensitive - material. This question can be answered with use of the point

kernel tabulated In Table 1.

Sample calculations of this problem have been done for the case that the

cell nucleus can be represented as a spherical source region with a diameter of

a few microns. The fraction A(d,E^) of the emitted energy absorbed in a sphere

of diameter d can be obtained as an integral over the point kernel. It can be

shown that

d

A(d,E ) = - ,Tl - 1.5 (r/d) + 0.5 (r/d)"^] F(r/r ,E )dr.

o r i L ' o o

o o

(2)

The fraction A(d,E^) is tabulated in Table 5 for source energies between 10 MeV

and 0.5 keV and for sphere diameters between 2 and 14 |j,m. If the source is

3
characterized by a spectruin S(E^) (continuous for beta-rays, discrete for conversion and Auger electrons) , then the average fraction of the energy absorbed

in the source region is

av o S(E ) E A(d,E )dE

o O O O (J)

where

E^ = S(E )E dE . av J

^ o o (4) o

is the average energy of the emitted particles.

The average absorbed fraction A (d) has recently been calculated by

av

19 / Ertl— for the radionuclides tritium and iodine-125. He assumed straight-line

motion of the electrons away from the point-isotropic source, and an effective

rate of energy deposition per unit pathlength in accordance with an experimental

range-energy relation. The relation used was one determined by Cole—''' in a

study of the penetration of low-energy electrons through collodion foils.

In the present work, the beta spectra and discrete electron spectra were

20/ 21/ taken from the work of Martin and Bl icher t-Tof t— and Gove and Martin, — except

19 / for iodine-125, whose discrete electron spectrum was taken from Ertl.— The

values of A (d) thus obtained are given in Table 6 for a few radionuclides. In av particular, a comparison is made with Ertl's results. It can be seen that agreement is close in the case of tritium, but not so good in the case of iodine-125.

4. REDUCTION FACTOR FOR A HALF- SPACE SOURCE

Another application of the tabulated point kernel has been made in order to

determine the depth-dose distribution in a (tissue-equivalent) water target

22 / exposed to electron and beta radiation from a surrounding radioactive cloud.

4
Such a cloud can arise as the result of the injection of reactor effluents into

the atmosphere, or as the result of a nuclear accident.

As shown in Ref. 22, the solution of this problem can be accomplished in

two steps: (a) First it is assumed that a radioactive source is distributed

uniformly through one half-space of a water medium, and one calculates the

depth-dose distribution in the other half space. (b) One then takes into account

that the radioactive source is distributed in air rather than water, and makes

the appropriatie corrections to account for the differences of the scattering

properties of the two media.

Step (a) involves the calculation of a half-space reduction factor for monoenergetic sources. Let R be the absorbed-dose rate that would prevail everywhere

o in an unbounded homogeneous medium if a uniform isotropic source emitting

electrons of energy were distributed throughout the entire medium. Let R(z,E^)

be the corresponding absorbed-dose rate at a deo th z>0 that would prevail if the

source were confined to the half-space z<0. The reduction factor , i.e. the

ratio G = ^^/R^, can be expressed as follows in terms of the scaled point-kernel:

G(^/^o'^o> -\\ ^1 - C^/O/t] FCt,E^)dt.

z/r

o

Values of the reduction factor for a water medium are given in the attached

Table 7.

5
References

1. L. V. Spencer, Energy Dissipation by Fast Electrons, N.B.S„ Monograph 1

(1959).

2. M. J. Berger, MIRD Pamphlet No. 7, Suppl, No. 5, J, Nucl . Medicine 12_, 5

(1971).

3. M. J, Berger, Monte Carlo Calculation of the Penetration and Diffusion of

Fast Charged Particles, pp. l35 - 215, Methods in Computational Physics,

Vol. 1, Academic Press, New York (1963).

4. M. J. Berger and S. M. Seltzer, Ann. N.Y. Acad. Sciences 161 , 8 (1969).

5. F. Rohrlich and B„ C. Carlson, Phys . Rev. 93, 38 (1954).

6. L. Landau, J. Phys. USSR, 8, 201 (1944).

7. 0. Blunck and S. Leisegang, Z. Physik 130, 500 (1950).

8. S. Goudsmit and J. L. Saunderson, Phys. Rev. _57_, 24 (1940).

9. N. F. Mott, Proc. Roy, Soc. A124, 475 (1929).

10. G. Moliere, Z„ Naturforsch. 2a, l33 (1947).

11. G. Moliere, Z. Naturforsch. 3a, 78 (1948).

12. M. J. Berger, pp. 157-174, Proc. Third Symposium on Microdos imetry , Oct. 18-

22, 1971, Euratom Document EUR 4810 d-f-e (1972).

13. A. E. S. Green and L. R, Peterson, J. Geophys . Res, (Space Phys.), 73^, 233

(1968).

14. L. R. Peterson and A. E. S. Green, J. Phys. B., 1, 1131 (1968).

15. A. Cole, Radiat. Research 28, 7 (1969).

16. International Commission on Radiation Units and Measurements (ICRU), Report

16, Linear Energy Transfer, Washington, D,C„ (1970).

17. L. E. Feinendegen, Tritium-labeled Molecules in Biology and Medicine,

Academic Press, New York (1967).

18. H. Ho Ertl, Li E„ Feinendegen and H. J. Heiniger, Phys. Med. Biol. 1_5, 447

(1970).

6
19. Ho Ho ErtL, Dosimetrie von inkorporiertem Tritium und Jod-125,

Kernforschungsanlage Julich, Report Jul-688-ME (1970)

„

20. M. Jo Martin and P. Ho Blicher t-Tof t , Nuclear Data 8, 1 (1970).

21. No B. Gove and M, J. Martin, Nuclear Data 1^, 205 (1971).

22. Mo Jo Berger, to appear in Health Physics.

7
...
/

E

O ' MeV

r/ro . n u 1.1 b . UUUb . u u n b . I) U 1 U .0015 . 0 , J 2 ^

. UU . / 1 i . o 7 -5 *bid . bU7 . 5 7 1 i .5o^

. Ub . . /'bb . 744 . 711 • b 7 4

. lu . 9UU . HbO . b2() . 770 . 74'!

. In i . Ur)f. 1 . Ulb .9b 4 .924 . Hb9 . b 3 b

,'d\J i . IMn 1.141) 1-. U74 1 . f)2b .^b3 .926

.^b i . sJbO i . ^tj4 1 . 1 <-:h) 1.142 1 . 0b7 1 . 0 2''

• ou i . 44c 1 . i99 1 .2b9 1.181 1 . 1^ "

. ^b L . r-) U _^ 1 .b^l 1 . 4bB 1 . 383 1 . 30 2 1 . ?5^

. 4U i . o6-j 1 .bU4 1 ,bdn 1 . 475 1 . 4 U 0 1 .35^-

.4b i . /'i:::^ 1 .bb^ 1 .bbb 1 .b39 1.474 1 .4 3b

. btJ i . /U^) 1 . 644 1 . b"b 1 .b53 1.520 1 .493

. bb i . b^b 1 .b9? 1 . bbb 1 . b43 1 .520 1.50^

« dU i . 4 o r-i I .46i 1 .474 1 .479 1 . 4 52 1 . 4bb

• Db i . 1 . 'dbd 1 . 3^4 1 .353 1 . 39b ] .41b

. /u . d / / . 99b 1 . lr?b 1 . 1 92 1 .254 1 .277

. /b . bJ!b . /bb . bbb . 9b2 1 .029 1 . Ob?

. HU . d! b '4 .44b . bOb . 885 .77o . 53^

. bb . U^JU .1/7 .bi;£ .403 . 50 9 . 59^

• U 1 . U4o .114 . Ib4 .309 .377

.yb . U U o . UUU . u n 0 . f)5U . 1 oO .210

1 . uu . U U L

< .UUU .UUU .(102 . 0 5 "1 . O'^O

I . Ub . UUU . 0 0 U . n tj 0 . U 0 C . OuO .020

t . 1 1 i • U U . U u t' nil f1 . linn 1 J u u n n Ti n .1 fi i . ib . UUU .UUU . nun .00 0 . OUO . 0 0 0

i . (dU . u u u . U U u • u f) 0 .000 .000 . 0 0^

r/r„ . uu^u . I)U4U . U !.i •• l~i . (lObU . f 1 n b 0 . n I fi n

. UU . b^j) .bcH . blH .514 . 511 . 50^

. ub * b d 'j) .31^ .bis . 59H . 590 . 5b «

. iU . no . b99 m n <b . 879 . 6o9 .bb^-

. i'J . /^o . 7 7b . /cc^ . 755 . 743 .73-^

. ciU . bb 'i . b'. 7 . B37 . bl7 . b0 3

. cdb . 9bc.^ . 94? .9:59 . 92 8 . ^Ob . b 9 U

. J)U i . U^i i . Uub 1 . I ' 4 3 1 . 0 3 U 1 . 0 L) b . 90

n

.^b I . ^ u b I . lo9 1 . 1 b4 1.129 1.10 3 1 . 0:V.

. 4U i . ^Ub 1 . d 7 ] L . 2 b !J 1 . 229 1.^07 1 . 1<1^ .4b 1 . ^9b 1 . :i 74 1 . 3b4 1 . 34 3 1.319 1 . 3

0'-'

» D'J i . 4oL, I . 44b 1.4,:. 3 1 .422 1.41 0 1 . 3 ^^

. bb i . 4^^^: 1 . 4a2 1 . M 7 7 1.47b 1 .474 1.47:-'

. bU i . 4 t-i o 1 . 4 b 's 1 .M ^ 9 1 .492 1 . 4 9>N 1 . 504

. b-i i . 4^:!^ 1 . 4^? 1 . 4 .5 7 1.44b 1 . 4o2 1 . 4 7b

. /U i . bU i 1 . it^ii I . 3 3b 1 . 351 1 . 37b 1 . 39"^

. /b i. Id::, 1 . Ibl 1 . 1 -.4 1.20 3 I .229 1.241

. :ju . ^Uu . 9br' . 9 1 . 0 0 0 1.02 b 1 . u 4

• b b ' bbci . 70 9 . 7:/3 . 757 . 7 b 3 .80^ ! . 1^ / .472 . M"4 .512 . 534 .54 p. .Mb . ^bl . i?bb .1' ''b .30 5 . 321 .32'^

1 . UvJ . l<:'u . 1 i9 . 1 . Ibl .173 .173

L . U •) . . J 4 H . Ubb . Ub?. .Ubb . Obb . 0 7 0

i . i U . U U J . Li 1 b . u.?u . f)2 2 . 0 2 4 .024

1 . ib .UUU . U tj U . UUU . (iO[; . 11 0 0 . uO^

1 . iiU . UUu . U U i 1 . 0 u 0 . 0 0 0 . 0 0 0 . 0 0 0

Table 1. Scaled point kernel F(r/r ,E ) in water.

o o
MeV

,

II 1 "-iL. . i 1 c. 1 j ' i .030(1 . 040 0 . 0 b 0 0 . 0 o 0 0

. V u • b U fi . bl ) . b 2 1 . b2F . b 3 o .64 4

. U .-) . b b f-' . b 7 o . b 7 9 . b b 2 . b :•) . b-''?

. 1 J • b b '4 . b H . b 4 3 .64 0 . 64 0 .64 0

» i : . / 1

/' . /ij b . C' ^"^ fl . f)9 0 . h90 . 6 b

'"^

. /b

. 7 -b -> . 7b4 .74 b .74 0 . 7 4 1^

. d b . dbV . b b b . b 3 4 . b2 (J . 1 4 . ;t 1

. y b / ,9^4- . '^\'>2 .92 0 . -U 2 . on.f-

. 3 b 1 . l)tDc_ ] . Ub') 1 . 0 3b 1.022 1 .110 1 . U 0

'^^

i . 1 /<d I . 1 o 2 1 . 1 b 1 1.14 5 1.14 0 \ .134

i . 2bi i . 2 7 1 1 . 2 M 1.29b 1.290 1 . 2 9

. bu I . 39 1 L . 3 b 1 . 4 ;3 1.417 1 .424 1.4;^ f •

. br^ i . 4 /b 1 . 4 ry 1 . b 1 ! :> 1 . b2b 1 . b b 0 1 . bS^

. b J 1 . bl J 1 . b 4 1

1 . b7b 1 . b07 1 . o 2 2 1 . 6 2:' .Ob L . b U 3 1 . b 2 M 1 . b 2 1 . b 0 g 1 .624 1 . b .5 4

i . 4 b 1 .449 1.4/9 1 . bO 1 1 .61'-) 1 . b 2

. /•:> L . 2 /'4 1.299 1 . 3 1 b 1 . 3 2 3 1.32 '.5 1.33

. b u i . U / "4 1 . U 9(1 1 . 0

'»0 1 . 0 b4 1 . 0 b 0 1 . 7 <^

. b D . b 1 y . i3 1 b . 7 9o . 7bU . 7 r5 b . 7 64

. y u . bbb . bb2 , b?3 . bOb .492 . 4 T 4

. S)b . 3 3 J . 33b . 3 L 3 . ?9U .^74 . d. b 4

1 . IJ u . 1 bb . Ibi . 1 u 4 .146 .134 . 1 ^b

i . u b . U /2 . U 73 . 0 74 . 072 . n 7 (3 . 0d6

1 . ij . U23 . U21 . 'J 1

.0 1b .Olb .01^.

i . ib . UUu . (J U U . 00 0 .000 . OOii . 0'"" '

. U iJ b . Lif)!] .11(10 .000 . 0 0 0 . Oon

r /r' o

. u bO ;j , 1 U 'J 1) . 1 b'Ol . ?oo(; . 3 0 U

'1 . 4 i'j 0

'1

. U J • bb9 . b72 . b :> b .oil .64 Of

. U . o I.I 3 .612 . b 3 3 . b 4 t* .674 . 6 9 n

. 1 J • b '4

. b 4 . nb 1 .678 . 7[1 ? . /2'-'

. i 'J . o9 1 . b92 . 7 0 3 .718 . 74^ . 7 6

. 2U . /m .74 4 . 7 b 7 . 770 .791 . 9i)7

• cib . bU9 . bUb . b2U . 8 2 6 .94b . bb6

.30 . 9U 1 . h99 . b'"-'*^^ .90 3 .

^' 3 1 2

. 3 b I . u U b i . U iJ 3 . 9(-;b . ^97 . ^^96 . 9?M

. 4 Ij i . 1 3 1 1.131 1.12 3 1.10b 1.113 1.10

. "4 b i . c:! y 4 1.29 b 1 . 2'^2 1.293 1.26 I . 2b6

. b'J 1 . 4^ 3 U 1.432 1 . ^4 2b 1.42b 1 . '4 0 b J . 39"

. bb 1 • bbii 1 . bol 1 . bb? 1 . b 3 b 1 . S22 1 . 60 7

. D I

1 . o 3 b 1 . c)4

1 . b4 3 t . n 3 9 1 .622 1 . n 0 4

. h b i . O 4 1 . b4 ^ 1 . b 3 1 . 6b 1 1 . 6 3

'3 1 .622

. 7U i . b ^ t.j 1 . b b 7 I . bb3 1 . ')bb 1.644 1 . b 5 4

. /b i . 3 4 2. 1.34 b I .334 1 . 3b9 1 . 3ol 1 . 3 6

• h U i . U ^3 1.073 1 . 0 73 1.07 2 1 . 0 7 1 1.071

. b b . /b / . /b2 .743 .74 0 . 7 3 b .73"

. y iJ . 4 o 4 . 4 o 1 .44 b .43b .4^9 . 4 2 .9b . d H o .23 7 . 2 1 .213 .193 . 1 b

1 . U 1) .114 . 1 u u . 0 " b .0 76 • ' J O . 0 r^ V

i . U b . U b b . U 42 . 0 2 u .020 .014 . ti 1 2

1 . i 0 . U 1 i . ! 1

f . . 0 0 7 . 0 0 b . 0 0 3 .

'1 0 2

i . lb . J 1 1 u .0 01) . 000 . 0 0 u . 0(10 .O'!0

i . rdU . J IJ u . 00!) . i! 1; 0 . 0 0 0 . 0 0 0 . ofin

4 *i

Table 1. Continued.
MeV

. bUDU . bUUU . bOtif) 1 . 0 0(10 1 . souo 2 . fi 0 0 n

. UU . b /4 . bB9 .714 . 73b .771 . 798

. Ub . /(JH . 7Z7 . 7'.)2 . 774 .811 . 83'i

. lU ,7^14 ,7b f . 7f^4 .807 .845 . 866

. lb . 7H4 . 79b . b20 .8 35 .871 . B9'l

. dlU . bidl . BbO . b67 . 897 . 920

. iib . 5bb . H 7 b . 892 .90 5 . 935 . 95?

. 919 . 924 .936 . 949 . 969 .980

.s5b .99b 1 . U H) 1.012 1.003 1.007 1.010

.40 1 . lUt+ i . U99 1 . Ubb 1 . 083 1.071 1.06?

. t+b i.a=4 7 1 .ii31 1.211 1 . 196 1 .157 1 . 13U

. bU i . 3B'4 1 . i 7 ;^ 1 . 355 1 . 338 1 .285 1 .23^

. bb 1 . 49B 1 . 4b4 1.457 1 . 435 1 . 380 1 . 325

. ou i.b79 1 .bbi 1 .529 1 .501 1 .4 38 1 . 393

.bb 1 . bUb i . bb7 1 .555 1 . 52b 1 .472 1 .42^

. /u 1 .bi^t^ 1 . 5 1 U 1 .4b9 1 .475 1 .44 0 1.41?

. /b 1 .s5bl 1 . i^b 1 . 339 1.334 1 . 319 1.313

. bU i . U /U 1 .Ub7 1 . Uo4 I .067 1.077 1.08 8

. bb . f5 ) . 7M2 . 74b .765 . 78?

.9U . .4^4 .427 .431 .455 .481

.yb . Ittb . ibJ5 . 1 b4 . 191 .206 .230

1 . Ul) . Ub« . IJbb . 0 5b .0 58 .066 .078

1 . Ub .UlU .UlU . UU9 . nob .003 .013

i . iU .OUd .001 .001 .001 .00?

i . lb . UUU , U U 0 . (1 0

0

. f)0 0 .0 00 .000

1 . U . u u u .UUU .00 0 . 0 0 0 . 0 (J 0 .000

r/r^ ,3. UUUU 4 . U U U U . OOIH) b . 0 0 0 0 8 . 0 0 u 0 1 Q . 0000

. UU * bdb . bbb .341 . 841 .8 39 . ^i 3 3

. Ub . bbo . Hba . H75 . 877 . 873 . 867

. lU . B91 . 9UU . 90 5 . 906 . '='03 . 897

.Id .91b . 9id4 . 929 . 930 . 925 .91"^

. ^U . 941 . 9bl) . 949 . 94b . ^41 .936

. 9b / .971.) .9^7 . 966 . 957 . Q4 8

. ^U . 9 ^9 .97b . 9 7b . 9 72 . '^b3 . 95^^

. ^b 1 . U U 7 1 . 0 U 2 . 99b . Q92 . c^BO . 964

. HU i . U 4 b 1 . Ui?b 1.010 1 . 0 n 0 . <^b6 . 9 76

. i+b L . U9b 1 . Ubb 1 . U '4 c. 1 . 0 32 1.010 . 98«

. bU 1 . Ibl 1.119 1 . 0 9tj 1.065 1 . 032 1.00 4

. bb 1 . i45 1 . 19i 1.149 1.119 1.066 1.024

. bO i . ilb 1 . 'db9 1 .2] 1 1.173 1 . 1 1 (3 1 . 0 54

. bb 1 . bbU 1 . J 1 .25 7 1.217 1.147 1 . 0 8<^

. /U 1 . bbb 1 . b 1 3 L .2 75 1 . 242 1.179 1.12 4

. /b 1 . ^9U 1 . if 6b 1 . 24Q 1 .22o 1 . Ibl 1 . 136

. HU 1 . 1 1 i i . 12b J . Irb 1.131 1.130 1.10 2

. Bb . 9UU . 945 . 962 . 986 . 99"

. yu . bb^ . b U . b54 . 694 . 756 .803

. 9b .311 . 352 . 380 . 465 . 54?

i . UU . 1 U4 . 129 . 155 . 181 . ^35 .295

1 . Ub . U'dU . U2b . 3 b . 045 .067 .098

.. 1vj1 f 1 s . (1 fl

s

. U l.J J . u \j 0 .01? • U X c_ . U c_ ' i . lb . U U 1 . U U 1 . f)(t2 .003 .006 . 0 0

0

1 .«iU .UUU . UU(3 . (J 0 0 .00 0 .001 . 0 0

Table 1. Continued.
E

0 r

0

(MeV)

"2. (g/cm )

( 1

< M ( I' c * c /cub

Ll A y f 1 >^ U i u

t 1 1 t 1 U U J J

1 ( 1 Q 1> 1 1

1 I , i 'J t \ u u CD 1 7 1 ' 1 — ( 1 t;

CI c J U

U U H U

U L) J U

U u D U 1 . ( 1 ^ 7 — I i Li

t J 1 1 K 1

*J L> U 1 . A U — 1 1 U. -i • C) O ^ U

i 1 1 1 1 1 1 L' i U U

U i D U D» U4ii~U4

u c u u

>i 7 Ll. — 1 1 Ll

c.i • . J / 4 — ij 4

U J) u u

1 7 1 — i i X U 4 U U

i 1 K 1 1 f 1 u r> u u

Li 5 'J J » 1 1 < U b u u

u n f i 7 » ! > X o • y U /

"• U J

> U O u u

» i u u u i • 4 I ) i U £1

> i D U U

J 7 1 1 » f 1 "D ^ f 1 1 1 1

» J U U U O • C O J I.' C

• 4 U C'l U

1 O / i — i"l 1 i • r>4 — i j 1

1 7 < ' \ ( 1 1

» o u u u . P p 7 — 1 1

• l5 U U U

11 » u u u u

Ll CI / — 11 1

11 » J 1 J U U c) • ^^DO u i

J

• Li U U U y • Q i J l' 1 <

.J

'

1 [ 1 1 1 1

1 tl U ^1 4 ( 1 (

J i • u ( ) Li U

1 O (.1 / + 1 ; i 1 D • U 0 i. 1 1.J C • V T tj U

b • i ) 1 1 U U y . LI j U U

lit . b u 0 U 4 . H B U + iJ 0

90 o yu

2 (g/cm ) 1.510-^5

. 6960b 2.016-06

. 7^"! 39 3 0 120-0 6

. 7 3969 4.420-06

• 7bbH3 8.358- 0 6

. 7 B 2 b li 1 . 338-05

. 79916 ?. . 620-05

. B 0 H 4 9 4 . 259-05

.8122b 6.215-Ob

. 81 bBl 8.160-0 5

. 8203b 1 . 386-04

.82328 2 .043-04

. 82b6B 4 . 1 b3-04

. 82482 6.907-04

.82016 1 . 4 07-03

.81626 2 . 327-03

. 81326 3 . 434-0 3

.81152 4,712-03

.80859 7.732-0 3

.81)760 1.131-02

.80416 2 . 219-0 2

. 80266 3. 532-02

.80049 6.614-02

.30011 1.011-01

. 30009 1 .388-01

. 8 Ll 0 4 3 1 . 783-01

.80 15b 2 .603-01

. 80343 . 4 b 2 - 0 1 .80731 5.616-01

.81334 7 .819-0

.82839 1 . 2 3 1) + 0 0

.84244 1 .68^+00

. 85551 2. 13 8 + 00

. 86458 2 . 5 M h f n f 1 .88849 3 . 5 1 (3 0 0

.91118 4 .447 + 0 0

Table 2. Electron c.s.d.a. range and 90-percentile distance in water.
F(r/r^, E^)

E =

o

1 MeV E = n 0.1 MeV

r/r

o

c. s, d. a. app rox

.

with

stragg.

c. s. d- a . approx.

wi th

scragg.

0.00 0.807 0,736 0, 588 0,572

0,05 0.814 0.774 0,605 0.612

oao 0.831 0,807 0.639 0.648

0.15 0.859 0.835 0,688 0.692

0.20 0.896 0.867 0.753 0.744

0.25 0.943 0.905 0.833 0,808

0.30 1.001 0.949 0.928 0,899

0.35 1.071 1 .003 1 .040 1 .003

0.40 1 .152 1 .083 1 . 166 1 .131

0.45 1 .247 1 .196 1 .305 1 . 296

0.50 1.353 1 .338 1 .456 1 ,432

0.55 1 .468 1 .435 1.610 1 . 561

0.60 1.573 1 . 501 1 .755 1 ,645

0.65 1 .636 1 . 529 1 .856 1 . 649

0.70 1.615 1 .475 1 ,833 1 , 557

0.75 1.450 1 .334 1.569 1 ,348

0.80 1.081 1 .067 1 . 049 1 ,073

0 85 \j . J cry C\ A Q/i U . / -)2 0.90 0.078 0.431 0,128 0,461

0.95 0.000 0.191 0,009 0,237

1.00 0.058 0.000 0.100

1 .10 0.001 0,009

Table 3. Comparison of point kernels for water calculated with inclusion of

energy-loss straggling, or in continuous-slowing-down approximation.
E

0

(stragg)/xgQ (c. s.d.a.

)

(MeV)

5 1 .077

J 1.067

2 1 .057

1 1 .055

0.5 1.055

0.3 1.056

0.2 1.058

0.1 1 .064

0.05 1.074

0.025 1.090

Table 4. Comparison of 90-percent ile distances in water computed with

inclusion of energy-loss straggling, or in continuousslowing-down approximation.
(VI

3

O >

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r~ CO ^ o

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c o o c; o o o o c o c o o o o o o o c o

a 0 c c c cr cr ir 3 _ _ ...

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r)XIXl3J^0333333333 3 3 0 0333300000333333 3 3 333-tr-(AJ-O3-r).0D3/>3333333 3 33333 3 3333 3 3333 3333333 3 333'-t-(?003-nX>-03 0333 3 3333 3 33333 3 3333333333333033333-t-4A|-0.-tr) OX 3^ 3 3 3 3 333

1—

f

CDU

•H

J-i

(1) a

en

c

OJ Xl

i-i c

(/)

CTi

(/) •H

4-J CD

4-1

;^

S-J 0)c

CD

CD

(J

o

<u 4J4J H

e

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0) ^ 0)

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cfl M-l •H O T3

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(1) co

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Cfl

!-i o

Pn w

in

(U

>-H J3

-I '-t Ai -o n .0

II
d E =5.68 keV E =49.3 keV E =48.8 keV E =3.34 keV

av av av av

(|j,m) Ertl This work This work This work Ertl This work

2 50.6 45.7 1.7 1.9 45.3 44.8

4 68,3 65,5 3„4 3.6 56.1 49 .7

6 76.0 75.6 4.9 5.1 63.1 53.0

8 81 .6 81 .4 6.4 6.6 68 3 56.3

10 84.9 85.0 7.8 8.1 71.5 59.5

12 87.5 87 .4 9.2 9.4 74.6 62.7

14 89.2 89.2 10.6 10.7 77.] 65.6

Table 6. Percentage of the emitted source energy, 100 A (d), that is absorbed

in a spherical source region of diameter d.
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^ O jD rO (T' -H CT' C\i a^ O JO r- cO AJ :± t3 'Al —» O O O 3 O O

3 3 -( r> 0^ n 3 —t o o .-t 0^—13333-333333

3 .'):^-^OjAJ-\I'-t-t-(333333333333 3 333

r) 3 O r) T -"i :1- J> -C :t "U T r^ .C -J 3 3 3 3 3 3 3 3

3 33::t-03J)'^r;r.'^-f3^3-nM3 333 3 333 -3

3 3.Tr^J-C—^DrMa^^C^>")—^-H33 3 333333333

3 .n-O'O'VAJr-*—1333 3 33333333 3 333 3 3

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o Or-(ro^^r*-rocj"o:±croor\ivo—•foa^'O.Ha'O—toooo r". c: lT »-i vT c" vc a, ^ r c^. f\ l.'' .-1 ^ f<~ c+ c c cl c: c c c 3 O—tiDCT'd-O^Cr03cr'^d-rOC\]—'0C3000C3 0C

o it; ^ ro <\' (V cvi «-i .-I ^ o o o c c c: C' c CT' c o o c <3 o 3

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OJ Or<OCClC\JfOTHK;o<\j'X!CO--iCC. C0'-t\r;'O--tGOOC333 O C.-t:^0^;J-OvCfOC(^.O^C\J^-H000 0 303CCO

3 Lr.:ffO(V'CV(\J^^^COOCOCCOOOC-3e03C

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o-^^cr;f^vDcGLni^pocJ-cr.JooJ>Lnv-"\j—13333 3 33

o ^ ^ cc ir oj <T IT fo cv T-i o c o 3 3 3 o 3 c c 0

lT Cv' C\l ^ ^ C 3 3 3 3' C. 3 3 3 3 O O 3 O 3 O C

u

o

3r^:OaOj-i^crr\jLrj-invrioja3-^^3- -0—133 3 3 CT rH rv- 0" ^- (V «-< rvj 1*0 —t 3 3 c o c 0 3rOCCrOaCcJ-—tCCvC'^r<)(V—(3 33 3 3333 lT. cf'OOoCVj^.rH^. 3 33 3 0 33303 3 3333 3

r: o

CC 3 fV- J*; cr 3 r'"! —I CT' (AJ -Xi J"! X fVI 3 3 3 3 3 3

3 3 .D lT, '.C 'Si — .-^^J —1 :r —( X 3 J; :v 3 3 3 3 3 3 3 3

3 3 3 r»- (\J or 3- —< :t —t 3 3 3 3 3 3 C 3 3 3 3 J^J-?OC\Jr\J^'H^033 3 33 3 33 r:03 3 33 3

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c L^1::^^Of^. CMCM>-<»H.-tooocoococoocrccoc;

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r<-;

c »- a. c LT ^ IT vC cv LT K, cr c3- cv if. •-i o cr c: c

c o ^ o 1^ o o cvj lO vO cr lt fv' c c o c; o c

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ro CM fv; ^ ^ »-i c: c o c cr o o o c: o C' cr o c. c cr

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lO J- f^; (\i (\i »-i —< ^ o c c c: c o cr c cr cr c: c c C' c

'j^ r CO vC' o a; ^ cf; CT' ro "-^ cr X c: cv <-H c o c; c o

-H c<-'Lr5c;vCrHX;s-.-icr'h-^rocv»^.-(Ococ:cooc?o

o o cr _r; o lO a- CT rv CD ^ ^ o vC o -1 o o o

cr c a: X (T' x; ^ C3 oo !^ X " r\j —f c; r-. c; fC c: c::; ^ o —< i^' o r-i r- .-T CT' !^ cv.: ^ -H c ri/ o -:3 ^ o o o

O J^.J-rOrjfM^J^r-t-HOOCDCrOOCrOCDOOOOOOO

3 co X) fO -T) c; —' D cJ- >o _ri CM —I r; 3 3 o 3 o

J^::l'0'0rM^'M-^'-^^33333 3333 333 3 333

3 3 rr a^ >- X) o n -o 3 r! -o M .0

.0 3^n^:J-Mi^^)^X'^j^r)3-jna^r)M

3 3-<i^3i^^'>^=^—tx)X):r"OrM.^333

3 ,P ::r ^ W :m —' <-( —I z: 3 3 3 3 3 3 3 3

'J n -t 3 3 3 3

—(3 3 3 3 3 3

3 3 3 3 3 3 3

3 r) 3

3 3

3 3 3 _n

-H M CM -O -O nf- :f 3 _n 3

n -D .0 n

r, 3

D J^ 3 n 3 r 3

3 3 -H ^ Al
o o o -3 'n o^ CO "O —( fxj 0

X) -D ^ -O (\) 'M A) —» '-I -H O 3

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0 333033303 0 0 33

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3 3 -( n a> -D -H :r -H jo a :f -o m -h 3 3 3 3 3 3 o 3 3 3

O r):r-0 vJ^vJM'-f--<-<333333333~ 333 3 3 3

c

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o c: K, vC- c\j cr vC .-I 3 O iT) O

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3 3 XJ —I :f J- '\J 3 r) .-^ O T' O .n .\j T

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3 n 3

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'MCM'0>0:J-cJ-X)X)i)X)'^ n 3 X) 3 X) 3 x: 3

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X 3

-I M

USCOMM-NBS-DC
U.i,. 3L^PT. OF CCMM. 1. Plj B L.IC A f K )\ OK KKPOKT NO. | -. Gov't Accei.si^Q

BIBLIOGRAPHIC DATA NBS IR 73-107 i SHEET 1 4. r!TLl£ AND SUBTITLE

iiupiovsa romt Kernels lor tieccron cinci DeLa-i\ay j-^ui> luie 7

5. l-'i.biic.itija D.;[s

6. P>-,r:'onnir;g Or^nization Cc^de

:;3 3 IR 73-107

7. althoPaS)

Martin J. Berger

a. i'errorniin? Organi nation 2400437

9. PERFORMIXG ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

Ut-PAKlMtNl Ur CUMMEKCh

WASHINGTON, D.C. 20234

10. pro)-cc/ rask/ *X'ork Ui:ir. No.

11. Con:ract/Grant No.

12. Sponsoring Orgcinization Name and Address

U. S. Atomic Energy Commission

Division of Biomedical and Environmental Research

Washington, D. C. 20545

13. Type of Report & Period

Covered

14. Sponsoring AgenC7 Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

A calculation has been made of the spatial distribution of absorbed dose in a water

m.edium around monoenergetic point-isotropic electron sources. The calculation takes

into account angular deflections and energy-loss straggling due to multiple Coulomb

scattering by atoms and orbital electrons; it also includes the transport of energy

by secondary brerass trahlung . The results are presented in the form of scaled point

kernels for 36 source energies between 10 MeV and 0.5 keV. Tne scaling is done by

expressing all distances in units of the electron mean range, and makes possible easy

interpolation to any source energy in the interval covered. In order to illustrate

the use of the point kernels, applications are made to t;70 problems arising in

beta-ray dosimetry. The first problem pertains to the self -absorption of energy in

spherical source regions. The other problem concerns the absorbed- dose distribution

as a function of the distance from a semi-infinite uniform source region.

17. KEY WORDS (Alphabetical order, separated by semicolons)

A.bsorbed dose; beta particles; depth-dose; dosimetry; electron; point kernel

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U5COMM-OC 5'5244-P71
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