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AlllOb D37233

MRCiD 32-2451

Tables of Energy-Deposition

Distributions in Water Phantoms

Irradiated by Point-Monodirectional

Electron Beams with Energies

from 1 to 60 MeV, and

Applications to Broad Beams

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Radiation Research

Washington, DC 20234

January 1 982

Prepared for Department of Energy

Office of Health and Environmental Research

Washington, DC 20545

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62-2451

1962

ce of Naval Research

igton, Virginia 22217

MATIORAX* BVKCAU

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NBSIR 82-2451

^ A TABLES OF ENERGY-DEPOSITION

DISTRIBUTIONS IN WATER PHANTOMS

IRRADIATED BY POINT-MONODIRECTIONAL

ELECTRON BEAMS WITH ENERGIES

FROM 1 TO 60 MeV. AND

APPLICATIONS TO BROAD BEAMS

MAR 4 1982

n6i ^ - ^

/16.1'r -^5

M. J. Berger and S. M. Seltzer

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Radiation Research

Washington, DC 20234

January 1 982

Prepared for Department of Energy

Office of Health and Environmental Research

Washington, DC 20545

Office of Naval Research

Arlington, Virginia 22217

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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TABLES OF ENERGY-DEPOSITION DISTRIBUTIONS IN WATER PHANTOMS IRRADIATED

BY POINT-MONODIRECTIONAL ELECTRON BEAMS WITH ENERGIES FROM 1 TO 60 MeV,

AND APPLICATIONS TO BROAD BEAMS

M. J. Berger and S. M. Seltzer

National Bureau of Standards

Washington, D.C. 20234

ABSTRACT

This report presents tables of elementary three-dimensional absorbeddose distributions in a water phantom irradiated by monoenergetic,

point-monodirectional electron beams. Such distributions have been

obtained by the Monte Carlo method for 14 beam energies from 1 MeV

to 60 MeV. The tabulated results can be applied to the determination

of absorbed-dose distributions from parallel beams of arbitrary finite

cross section. The beam of interest is treated as a superposition of

point-monodirectional beams, and the absorbed-dose distribution is

obtained as a corresponding superposition of elementary absorbed-dose

distributions. By way of example, the tabulated data are used to

obtain (1) depth-dose curves and practical ranges in broad-beam

geometry, and (2) central -axis depth-dose curves, radial distributions

of absorbed dose, and isodose patterns for beams with finite circular

cross section.

Key Words: Dosimetry; electrons; Monte Carlo; point-monodirectional

beams; superposition; treatment planning.

This report is based in part on a paper presented at the Electron

Dosimetry and Arc Therapy Symposium, Wisconsin Clinical Cancer Center,

University of Wisconsin, Madison, Wisconsin, September 10-11, 1981.
1. INTRODUCTION

The determination of absorbed-dose distributions from electron beams, with the

accuracy required by treatment planning, is a complex problem whose solution is

facilitated when experimental dosimetry is supported by calculations. Flexible,

approximate, analytical methods are required as well as detailed transport

calculations. Among the latter, the Monte Carlo method (track simulation) is

probably the most expeditious if not the most economical approach, because it

requires the fewest assumptions and approximations and allows the fullest

utilization of the available cross sections for the interactions of electrons

with matter. The Monte Carlo calculations fall into two categories. The first

category consists of special-purpose calculations that simulate as closely as feasible the beam and medium configurations encountered in practice. The second

category consists of calculations for schematized and somewhat oversimplified

configurations, and is useful for general surveys and parametric studies; it

also can provide benchmark results for testing experimental dosimetry methods,

and may be useful for checking the validity of assumptions made in approximate

analytical treatments.

The present paper describes Monte Carlo calculations that fall into the second

category. Attention is focussed on the application of the superposition method

[l-4y to the calculation of absorbed-dose distributions in a homogeneous water

phantom. According to this method, the electron beam of interest is treated as

a superposition of point-monodirectional beams, and the spatial distribution of

absorbed dose is obtained by a corresponding summation over the elementary

absorbed-dose distributions from point-monodirectional beams.

Such elementary distributions have been obtained for monoenergetic pointmonodirectional beams with energies from 1 MeV to 60 MeV. Some of the characteristics of these elementary distributions are described here, and examples are given of the application of these results to other beam geometries.

2. ELEMENTARY ABSORBED-DOSE DISTRIBUTIONS

2.1. Definitions . The phantom is assumed to be a homogeneous semi -infinite

medium that occupies the region z > 0. A point-monodirectional (i.e., a narrow

pencil) beam of electrons with kinetic energy To, traveling along the z-axis,

is assumed to be incident perpendicularly^ onto the surface of the phantom. The

Monte Carlo method has been used to obtain the elementary distribution function

F(p,z), defined such that F(p,z)dpdz represents the average fraction of the

incident beam energy that is deposited at depths between z and z + dz and at

radial distances from the z-axis between p and p + dp. The absorbed dose at

a point (p,z) is given by (To/d) F(p,z)/2ttp, where d is the density of the

medium.

^Figures in brackets indicate literature references at the end of this paper.

^The assumption of perpendicular incidence is appropriate for electron beams that

have been broadened by scanning, but does not take into account the angular

dispersion of a beam broadened by a scattering foil. In principle it is possible

to consider elementary point-monodirectional beams incident at various angles,

and to treat the angular dispersion by an appropriate summation over such

elementary beams.

2
2.2. Method. The Monte Carlo model, which has been described earlier [5-7],

takes into account the transport of energy by the primary electrons from the

beam, and by all generations of secondary electrons and photons. The transport

of electrons is simulated by dividing their tracks into many small segments, and

sampling the multiple-scattering angular distribution in each successive segment

from the Goudsmit-Saunderson distribution and the energy loss from the Landau

distribution. The histories of bremsstrahlung photons emitted by the electrons

are followed by sampling individual Compton-scattering , pair-production and

photo-electric absorption events. The electrons set in motion in these interactions, and the secondary electrons resulting in ionization events, are treated

in the same manner as the primary electrons. The energy deposition along the

electron tracks is scored and classified to obtain the distribution F(p,z).

2.3 Ranges . When graphing or tabulating elementary absorbed-dose distributions,

or other distributions derived from them, one finds it useful to express the

distances p and z in units of the CSDA range^ ro. When this is done, the

distributions corresponding to different beam energies To "scale" in the sense

that the explicit dependence on To is minimized, which in turn facilitates

interpolation with respect to To.

In the treatment of energy loss by ionization and excitation in the Monte Carlo

calculations whose results are reported here, the stopping properties of water

were characterized by a mean excitation energy I = 65.1 eV, and the density

effect (reduction of stopping power due to the polarization of the medium) was evaluated according to the prescription of Sternheimer and Peierls [8]. In

recent work on stopping power we were led to the conclusion that improved

accuracy would be obtained with a mean energy I = 75.0 eV, and with the density

effect correction derived by Ashley [9] from experimental optical data for water.

The old ranges that correspond to the cross sections used in the Monte Carlo

calculations are compared in Table 1 with the new ranges that would result from

the use of a mean excitation energy I = 75.0 eV and Ashley's density-effect

calculations. The new ranges are larger than the old ranges by 0.3% at 60 MeV,

1.2% at 10 MeV and 1.7% at 1 MeV. Inasmuch as these differences are small, it

does not appear necessary to repeat the Monte Carlo calculations with the

improved cross-section input data. It seems adequate to assume that the absorbed

dose distributions, as functions of p/ro and z/ro, remain unchanged, so that

it is merely necessary to use the appropriate new range values when expressing

the absorbed-dose distributions as functions of p and z.

2.4. Elementary Absorbed-Dose Distributions from Point-Monodirectional Beams . A typical elementary distribution of absorbed energy from a 20-MeV pointmonodirectional beam, is shown in Fig. 1. The information is equivalent to that

contained in the distribution F(p,z) as defined above, but the representation

is slightly modified. The histogram in the upper right-hand corner is a depthdose curve in the form of a plot of the dimensionless quantity (ro/To)^ vs. the scaled depth z/ro, where 3) is the energy deposition per unit depth

^The CSDA range ro, evaluated in the continuous-slowing-down approximation,

represents a rectified path length over which the eTectron kinetic energy is

reduced from To to zero. It is calculated as an integral with respect to

energy over the reciprocal of the total stopping power.

3
(obtained by integrating F(p,z) over all p). The other histograms show, for

the six indicated (shaded) depth intervals, the radial distribution of deposited

energy, as a function of the scaled distance p/ro from the z-axis. Each radial

histogram bin represents the fraction of the energy deposited in the indicated

ring around the z-axis. Each radial histogram is normalized to unit area.

Figure 2 gives elementary absorbed-dose distributions in a different form, in

terms of the normalized cumulative distribution

For point-monodirectional beams with energies of 5, 10, 20, and 40 MeV, contours

are given as functions of the scaled depth z/ro along which has the value of

20, 40, 60, 80, 90, or 95%. For example, the curve of $90 vs. z/ro indicates

the radial distance from the z-axis within which 90% of the energy is deposited

at each depth.

Figure 3 shows plots of ^n(l - 4’) vs. (p/ro)^, at various depths in a medium

irradiated with 10- and 20-MeV point-monodirectional beams. The quantity 1 - ^ represents the fraction of the energy at a given depth that is deposited at

radial distances greater than p/ro. If the radial distribution of deposited

energy were Gaussian, an approximation made in certain approximate dosimetry

calculations, then the curves in Fig. 3 would be straight lines. This is not the

case, but a straight-line approximation might be adequate over the radial region

around the z-axis in which ~ 3/4 of the energy is absorbed.

2.5. Tables of Elementary Absorbed-Dose Distributions . Monte Carlo results for

the distribution F(p,z) are given in Tables A1 and A2 at the end of this paper for

14 beam energies between 1 MeV and 60 MeV. At each energy, the spatial distribution of deposited energy was obtained by analyzing the sampled electron-photon

cascade from 10,000 incident electrons. Energy deposition was scored in annular

volume elements. The depth and radial boundaries of these volume elements were expressed as fractions of the CSDA range ro , and the fractions were kept the same for each beam energy in order to take advantage of the scaling of the

distributions.

Table A1 gives the distribution as a function of the depth only.

where = n ro/20, n = 1,2,... 20. It should be noted that the quantity

To Hp,/(Zp - Zp_i ) is an approximation to 100<Z)(z), where 3){z) is the energy

deposition per unit depth at z = (z^ + Zp_i)/2.

Table A2 gives the normalized cumulative radial distribution

00

(

1 )

( 2 )

4
f„(p) = 100 (3)

dp'FCp'.z)

dp'F(p',z)

5

which is an approximation to the quantity 100 $(p,z) defined by Eq (1).

3. SUPERPOSITION METHOD

Let the incident electron beam be characterized by an intensity distribution

Qo(xo,yo)» such that Qo (xo ,yo )dxodyo represents the number of point-monodirectional beams incident perpendicularly onto the surface of the phantom at lateral

positions between Xo and Xo + dxo and between yo and yo + dyo. Let

D(x,y,z) be the absorbed dose at the point (x,y,z), obtained by summing over

the absorbed-dose distributions from the point-monodirectional beams. To

facilitate the summation, we replace the coordinates Xo, yo by cylindrical

coordinates p,ip, such that

Qo(xo,yo)dxodyo = Qo(x + pcosv>,y + psinv’)pdpd»^ = Q(p,<^ )pdpd</j . (4)

The absorbed dose can then be written as

2

D(x,y,z) = /

Pdp J*

dv? Q(p,</j) ^2wp^

'' * (5)

0 '0

It is useful to introduce the beam-geometry function

•2tt

'0

1 G(x,y,p) = J

d'P Q(p,</’) , ( 6 )

in terms of which the expression for the absorbed dose can be re-written in the

form

D(x,y,z) = I G(x,y,p) F(p,z)dp . (7)

We note that the absorbed dose is an integral over two factors: one of them,

G, depends only on the beam geometry relative to the point (x,y,z); the other,

F, is the elementary energy-deposition function available from our Monte Carlo

calculations. Of particular importance is the uniform beam of finite cross

section, characterized by a beam intensity distribution Q that is constant over

a certain field size, and zero elsewhere. This will be discussed in Section 5.

5
4. BROAD-BEAM DERTH-DOSE DISTRIBUTIONS

We now consider the case of a uniform broad beam of unlimited lateral extent,

i.e., a beam with intensity distribution Q{p,</’) = constant. Depth-dose distributions from such beams (for beam energies between 1 MeV to 60 MeV) are shown in

Fig. 4 and are given numerically in Table 2. These plots were obtained by drawing

smooth curves through Monte-Carlo-generated histograms similar to the histogram

for a 20-MeV beam shown in the upper right-hand corner of Fig. 1. The depth-dose

curves are given in scaled form, as plots of the dimensionless quantity (ro/To)^

vs. the scaled depth z/ro, with 3) again the energy deposited per unit

depth. The curves extend beyond the depth z = ro because of the combined

effect of energy-loss straggling and the transport of energy by secondary

bremsstrahlung photons. The higher the beam energy, the more noticeable is the

bremsstrahlung tail of the depth dose curves, which is due to radiation produced

by electrons within the phantom.

It is customary to obtain the practical range by fitting a straight line to a portion of the depth-dose curve beyond the peak. As illustrated in Fig. 5, one

can extend the straight line to the depth where the extrapolated dose is zero,

obtaining the practical range rp; or one can extend the line to the depth

where it meets the bremsstrahlung tail, obtaining the practical range rS. We

have analyzed the depth-dose curves in Fig. 4 in this manner, obtaining first

the ratios Tp/ro and r^/ro at several energies, and then the practical ranges

rp and rg with the use^of appropriate ro-values. Linear least squares fits were made of the ratios ^p/To and rJ/To as functions of 1/To, which lead to

the following relations (with rp and^ r^ in cm of H 2 O, and To in MeV):

r = 0.501 To - 0.110 ) r I

) using "old" range values ro from Table 1 ( 8 ) r* = 0.496 To - 0.104

r /

r - 0.505 To - O.lOo)

' I> using "new" range values ro from Table 1 (9)

r* = 0.500 To - 0.101

It is estimated that the practical ranges have an overall uncertainty of ~ 2%.

In a further analysis of the depth-dose curves in Fig. 4, we have determined the

depths Z 50 at which the curves have fallen to 50% of their peak value. The

relationship between Zso and the beam energy To is almost, but not quite,

linear. In Fig. 6 we give the ratio T 0 /Z 50 as a function of Zso* and the

reciprocal ratio Z 50 /T 0 as a function of To, from which one can determine

To as a function of Z 50 and vice versa. Two sets of curves are given in

Fig. 6 , one (solid curves) obtained with the "new" ranges, and the other (dashed

curves) with the old ranges from Table 1.

6
5. DOSE DISTRIBUTIONS FROM BEAMS WITH CIRCULAR CROSS SECTIONS

The geometry factor G given by Eq (6) can be evaluated numerically when the

beam intensity Q is variable, and often analytically when Q has a constant

value Q over a prescribed "field size" and vanishes elsewhere. A particularly

simple cSse is that of a circular field of radius R, for which we shall give

some illustrative results. Let D(p,z) denote the absorbed dose at a depth z in the phantom and at a radial distance p from the axis of the incident beam.

The expression** for the absorbed dose, obtained by summing over a distribution

of point-monodirectional beams, is

00 D(p,z) - dp' G(p,p‘) F(p',z) . (10)

By some simple geometric considerations it can be shown that G(p,p') is given

by the following equations:

p

= 0

If 0 < p < R

R

R

If p

= R

If p > R

P

P

0 < p' < R

p > R

0 < p' < R - p

p < p' < R + p

p < p'

0 < p' < 2R

2R < p'

0 < p' < p

- R

R < p' < p + R

R < p'

G- Q,

G = 0

G = Q,

G

= (Q^/tt) cos

G = 0

G

= (Q^/tt) cos

G = 0

G = 0

G = (Q^/tt) cos

G = 0

[{p'^+p^-R^)/2p'p]

(p72R)

£(p'^+p^-R^)/2p'pJ

(lla)

(lib)

(11c)

(lid)

Figure 7 shows central-axis depth-dose curves D(0,z) calculated from Eqs (10)

and (lla) for beams with circular cross sections. A series of depth-dose curves

are plotted for a set of increasing beam radii (field sizes), illustrating the

transition from narrow-beam to broad-beam conditions. The dependence of the

shapes of these curves on the field size can be understood in terms of the

competing effects of electron multiple-scattering into and out of the central

axis region. Whereas the outscattering is independent of the field size, the

**This expression is equivalent to Eq (7); however, x and y are replaced by

p, and p is replaced by p'.

7
inscattering increases with field size, which results in a larger buildup and

peak and a slower decline beyond the peak for the broad-beam depth-dose curves.

Figure 8 shows the radial distribution of absorbed dose at various depths, for a 20-MeV beam with a radius of 1.25 cm, obtained by application of Eqs (10) and

(1 la-1 Id). The radial distributions actually are averaged over small depth

intervals. At shallow depths, the radial distributions are almost constant over

the irradiated field, and then drop off very rapidly. With increasing depth the

drop-off becomes less sharp, due to the effect of multiple scattering. The

radial distributions can be used to generate conventional isodose contours.

Figure 9 shows isodose patterns derived in this manner for 10- and 20-MeV beams

with radii of 2,5 and 5 cm.

6. CONCLUDING REMARKS

The results in this paper are only illustrative. The body of information con- tained in the calculated absorbed-dose distributions at 14 energies from 1 to

60 MeV remains to be exploited more fully. A very simple and straightforward

extension would be to compute depth-dose curves and isodose curves for fields of

various sizes and shapes, including beams with distributed energy spectra. A more difficult extension would be to discard the assumption of perpendicular

incidence and to take into account the angular dispersion of the incident beam.

Finally, it remains to be investigated to what extent the superposition method

can be retained when treating inhomogeneous phantoms.

ACKNOWLEDGEMENT

This work was supported by the Department of Energy and by the Office of Naval

Research.

8
REFERENCES

1. J. R. Clarkson, Br. J. Radiol 14, 265 (1941).

2. W. J. Meredith and G. J. Neary, Br. J. Radiol 1_7, 75 (1944).

3. S. C. Lillicrap, P. Wilson, and J. W. Boag, Phys. Med. Biol. 30 (1975).

4. K. R. Hogstrom, M. D. Mills, and P. R. Almond, Phys. Med. Biol. 445

(1981).

5. M. J. Berger, Methods in Computational Physics, Vol . 1, pp. 135-215,

Academic Press, N.Y. 1963.

6. M. J. Berger and S. M. Seltzer, Ann. N.Y. Acad, of Sci . 161 , 8 (1969).

7. M. J. Berger and S. M. Seltzer, Phys. Rev. C 2 , 621 (1970).

8. R. M. Sternheimer and R. F. Peierls, Phys. Rev. B3, 3681 (1971).

9. J. Ashley, Radiat. Res., in press.
TABLE 1

CSDA Range in Water

To

(MeV)

'^0 (g/cm^) Percent

Difference

01 d"^ New^

60 22.65 22.72 0.3

50 19.72 19.79 0.4

40 16.53 16.62 0.5

30 13.05 13.14 0.7

20 9.209 9.295 0.9

15 7.122 7.200 1.1

10 4.902 4.963 1.2

8 3.969 4.021 1.3

6 3.006 3.048 1.4

5 2.512 2.547 1.4

4 2.008 2.037 1.4

3 1.493 1.514 1.4

2 0.9654 0.9802 1.5

1 0.4305 0.4377 1.7

Old: Mean excitation energy 65.1 eV; density effect correction

O from Sternheimer and Peierls.

New: Mean excitation energy 75 eV; density effect correction

9 from Ashley.
TABLE 2

Scaled distribution of deposited energy per unit depth, for broad parallel

monoenergetic beams. To obtain the absorbed dose in units of_qrays (J/kg),

the entries in this table should be multiplied by 1.602 x 10~^° N To/ro,

where N is the number of electrons incident per cm^, To is the beam

energy in MeV, and ro is the electron range in g/cm^.

N O

1.0 1.5

Electron Beam Energy, T^

2.0 3.0 4.0

, MeV

5.0 6.0 8.0

0.00 .747 .794 .815 .836 .846 .850 .843 .837

0.05 .861 .884 .896 .905 .907 .902 .901 .892

0.10 .988 .984 .980 .970 .961 .955 .949 .937 ' 0.15 1.115 1.090 1.071 1.044 1.024 1.008 .995 .974 ,

i 0.20 1.239 1.190 1.156 1.115 1.085 1.062 1.043 1.015

0.25 1.352 1.287 1.249 1.195 1.155 1.122 1.094 1.050 i 0.30 1.446 1.389 1.344 1.280 1.231 1.189 1.154 1.098

0.35 1.524 1.481 1.441 1.375 1.317 1.260 1.209 1.139

0.40 1.559 1.516 1.480 1.418 1.365 1.320 1.261 1.178

0.45 1.543 1.527 1.510 1.450 1.405 1.365 1.326 1.242

0.50 1.485 1.482 1.472 1.446 1.415 1.386 1.350 1.274

0.55 1.393 1.396 1.397 1.391 1.384 1.375 1.360 1.302

0.60 1.259 1.288 1.302 1.314 1.320 1.322 1.314 1.280

0.65 1.073 1.115 1.143 1.180 1.203 1.220 1.230 1.228

0.70 .868 .928 .967 1.020 1.055 1.080 1.103 1.130

0.75 .640 .700 .742 .810 .860 .901 .935 .995

0.80 .445 .500 .540 .604 .660 .706 .750 .825 j 0.85 .275 .315 .347 .403 .452 .497 .540 .618 ! 0.90 .135 .158 .178 .217 .252 .290 .325 .400 ! 0.95 .0515 .0610 .0700 .0900 .109 .131 .154 .210

1.00 .0158 .0191 .0223 .0287 .0360 .0440 .0550 .0895

1 .05 .00450 .00530 .00600 .00745 .00910 .0113 .0150 .0250 1 1.10 .00103 .00127 .00150 .00194 .00250 .00320 .00450 .00850

1.15

) .00029 .00041 .00054 .00089 .00135 .00200 .00275 .00460

, 1.20 .00020 .00032 .00043 .00072 .00110 .00153 .00206 .00339

1.25 .00016 .00026 .00036 .00062 .00096 .00132 .00183 .00311

1.30 .00013 .00022 .00031 .00055 .00084 .00115 .00165 .00285

1.35 .00011 .00018 .00026 .00048 .00074 .00102 .00150 .00255

1.40 .00009 .00015 .00023 .00042 .00056 .00094 .00133 .00225
TABLE 2 (continued)

10 12

Electron Beam Energy,

15 20 30

, MeV

40 50 60

0.00 .826 .820 .810 .793 .750 .708 .683 .652

0.05 .882 .875 .858 .839 .797 .761 .733 .707

0.10 .926 .910 .894 .871 .831 .799 .767 .743

0.15 .952 .940 .924 .899 .855 .822 .791 .764

0.20 .993 .975 .949 .920 .872 .838 .804 .777

0.25 1.023 .997 .969 .938 .886 .848 .815 .786

0.30 1.053 1.028 .990 .953 .897 .856 .822 .792

0.35 1.087 1.054 1.011 .970 .906 .861 .828 .798

0.40 1.123 1.081 1.033 .983 .913 .866 .830 .801

0.45 1.162 1.111 1.059 1.000 .920 .870 .831 .803

0.50 1.200 1.145 1.086 1.017 .926 .870 .830 .802

0.55 1.226 1.169 1.108 1.035 .930 .868 .828 .799

0.60 1.233 1.180 1.120 1.050 .931 .863 .820 .789

0.65 1.209 1.178 1.121 1.053 .930 .854 .809 .777

0.70 1.145 1.130 1.101 1.045 .921 .891 .794 .760

0.75 1.040 1.062 1.052 1.006 .904 .826 .775 .741

0.80 .897 .946 .966 .937 .877 .801 .750 .720

0.85 .685 .765 .825 .840 .826 .767 .718 .689

0.90 .475 .550 .625 .686 .740 .706 .669 .648

0.95 .275 .340 .413 .500 .610 .620 .603 .597

1.00 .129 .165 .215 .310 .450 .508 .525 .525

1.05 .0400 .0610 .0910 .155 .293 .380 .430 .443

1.10 .0130 .0195 .0343 .0665 .162 .256 .320 .352

1.15 .00670 .01 050 .0159 .0275 .078 .151 .220 .254

1 .20 .00500 .00700 .0107 .0185 .0425 .0850 .143 .188

1.25 .00465 .00650 .0098 .0160 .0337 .0543 .0950 .1300

1.30 . 00435 .00605 .00900 .0146 .0290 .0467 .0657 .0890

1 .35 .00400 .00557 .00845 .0139 .0266 .0438 .0550 .0673

1 .40 .00355 .00520 .00800 .0135 .0258 .0410 .0525 .0610
FIGURE CAPTIONS

Fig. 1. Distribution of energy deposited by a 20-MeV point-monodirectional beam.

Fig. 2. Curves of scaled radial distance vs. depth, along which the normalized

cumulative radial distribution of deposited energy is constant.

Fig. 3. Fraction of energy deposited at radial distances greater than p/ro

a. 10-MeV point-monodirectional beam

b. 20-MeV point-monodirectional beam

Fig. 4. Scaled broad-beam depth-dose curves.

Fig. 5. Definition of practical range.

Fig. 6. Relation between incident beam energy To and depth Zso at which the

broad-beam depth-dose curve has fallen to 50% of its peak value.

Fig. 7. Central-axis depth-dose curves from beams with circular cross section

(radius R).

a. Beam energy 10 MeV

b. Beam energy 20 MeV.

Fig. 8. Radial distribution of absorbed dose from a 20-MeV circular beam with

radius R = 1.25 cm. Curves shown are averages over indicated depth

intervals.

Fig. 9. Isodose curves in a water phantom irradiated with 10- and 20-MeV

electron beams with circular field sizes.
RELATIVE

RADIAL

DISTRIBUTION

WATER PHANTOI^

yo/ro

Fig. 1
WATER

PHANTOM

or
0. 55-

0.60
0. 75-

0.80

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Fig. 6
CENTRAL-AXIS

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Fig. 7a
CENTRALAXIS

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Fig. 7b
Fig. 8
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TABLE

A2.

Normalized

cumulative

radial

distribution

of

deposited

energy.

The

numbers

in

the

column

headed

""Scaled

radius"

represent

the

scaled

radial

distance

from

the

< E

3: cu •o

p— cu >i 4-> CU

fO ro E

h- o CU a •r— a) T3 s •1

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cn 4-> r— E to

•1

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ro

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ro o

(A CL

(/) <U O

O) cu X5 4-> •r—

+-> +-> >1 r—

•r— cn ro

dJ +-> E E

> E CU CT

ro ro E cu -E E CU cr to = E •1— (A cu ro 1— .E cu ro 4U JD > Cl E CU 4-> CU E E V/

+-> ro CU E -o

•r“ r

•1“ to

ro o cu JZ > E o +J E • r- E

CL cu ro

CU CU +-> XJ E JC to

•1+u •1

o o CU JZ <+-

1 +J o 1

ro Q. ro o CU cu •r- cn •a cn X3 ~

ro ro • .E +-> E r—

cu U E < JZ ro CU +-> h- cu o ro cu E r— E CU PI JZt

• O CL 1

ro o M- ro H- E CU > — to .E E E

CU 1— CU •r—

•1+J E E E

»* -M • •r— CU

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ro cu CL

1 JZ CT CU to N h- LU -o *1—
Cumulative

Radial

Energy-Deposition

Distribution

in

Water*

Beam

Energy

=

60

MeV

ITl o CO o 00 00 O' V— O >d- in <7s to Sd- to o in to o to to CO to O CO O so PJ O PJ sd* vd- vd- vd- o in O' o 'd“ 00 PJ PJ m n- O' O

m vO •«d- O' to sO o to 'J- o 'cr — to h'. PJ sj* so O' in so PJ vd- *d" PJ SO PJ O m O PJ in n- O' o *- PJ K) sd- in in in p> n- 00 00 O' O' CT' CT' O

o O o so r— 00 OS 00 so to 00 PJ in 00 O PO to >d- in in vO h- 00 CO CO CO 00 O' c^ O' O' CT' CT' O' CT* CT* c^ CT* c^ O' CT' O' O' O' O

PO PO to to sd“ in sO r-' CO 00 CO O' O' O' O' O' os O' O' O' CT' O' CS O' O' O' O' O' CT. C^ CT' O' CTs O' CT' O' CT' O' O' CTv CT' CS O

o in o in O' >d- to o PO in so PO sd- m o SO in to 00 CT' in 00 in sd* sO PJ to vd- PJ in PJ CO in o to 00 to m sO O O

iTl m n- o to o in o so so o in — O' >d- sO m C?' O' so •- SO — «d- CO PJ in O' o — PJ lO lO 'd' *d- in •o so so r>. 00 oO CT' CT' c^ O' c^ O

O o o (M ^£> o in r>. sd" O sO 00 in o m CO o PJ to 'd' in sO vO r-. CO 00 CO CO CT' O' CT' CT' CT* O' CT' O' O' CTS CTS CT* CT* O' C^ O' O' CT' O' O

CO CO to sj- sd- in so CO 00 00 O' O' O' O' O' O' O' O' O' O' O' C7' O' CT' CT' O' CT' CT* C^ O' CT' O' O' O' CT* CTS CT* CT* CT' O' O' CTv CT' o

in o CO 00 <7' o to in 00 00 so O' in O' in CM 00 in 00 'd- vd- in m in O' o vO lO O vd" 00 o to SO PJ CO sd- 00 to vd- r'. c^ o

<d- m *- O' so 00 o ':P O' PJ m CO PO PO to 00 O' to n- — sd" hv C7V PJ in r-. O' o PJ to to vd- in m in sO so SO p. 00 00 OS O' CT* c^ OS o

o o to 00 O' so to P-. 00 sO PO p' o PJ to sd* in so sO rv. r'- CO eo CO CO O' O' O' CT' CT' O' O' O' CT' O' CTv O' O' O' O' C^ OS CT' O' O' o

CO CO to scr in in so 00 00 O' O' O' O' O' O' os O' os O' O' O' O' O' O' CT' O' O' O' O' O' c^ OS CT' CT' O' O' O' o CT' O' O' O' CT' o

o in o in so O' r-- PJ (M o PO 00 00 PO hv 00 00 r-v 00 O' in in to in o PJ so SO o 'd- 00 sd* vD CT' m o to so 00 in r'. 00 C^ o

'cr K) to o o Pv) in 00 O' PO PO to *- 00 o O' sO ,r- in 00 o CM vd- so CO o *- to sd- sd* in in m so so 'O sO 00 00 00 eo C^ CTv O' O' OS o

o o in o c^ 00 Cv IT) PJ OS 00 in O' PO to in in so hv 00 CO 00 CO 00 CTs ex' c^ O' CT' cs O' CT* CTs O' O* O' O' CT' CT* O' CT' CT' c^ O' CTv O' o

•— PJ Pd to sT in so sO 00 00 O' c^ O' O' O' O' O' os OS O' O' CJs O' O' CT' CTS O' CT' CT* O' CT' O' O' O' CTv CT* O' O' CTv O' O' O' O' O' CT' CT' o

m o sd- to PJ c^ to CO 00 00 vd- vd- C^ sd- Cs to PJ CN 00 CO in PJ lO 00 to to SD PJ in rO p. PJ SO CN «— to in eo Os O PJ in O PJ sd- sO 00 CN o

to *d- m — n- to PJ CTS o SO vd- O o vd- o CN m — in CN PJ sd- so CN O to sd- in in sO sO sO r*. n*. CO 00 eo O' CN CN O' Os CN O' o

o o PJ 00 CO 00 o O CTs in o p^ to in in sO h- r-v. 00 00 00 CO 00 OS CN CN CN CN CN CN CN CN CN CN CN CN OS O' CN CN CN CN CN CN CN OS CN OS o

iH PJ sd- in SO SO OO CO CT' Os CTV CN CN CN CN CN CN O' CN CN CN CN CN CN CN CN CN CN CN CN CN Os CN CN CN CN CN CN OS CN CN CN CN CN CN CN o

M o in o o in PJ o PJ sO so o PJ CT' o in vd- 00 sD vd- SO vd- PJ vd- eo n- to vd- N*. CN to sO sO p- CN O in r** O' to in sO sO O' o

+> to to to 00 00 00 00 sd- 00 •- PJ CO r>- o 00 sd- 00 ro in r>- 00 o — PJ to to sd- in SO sO SO SO sO r-- 1-^ n* p~ t-v eo 00 00 eo CO CN OS CN CN CN CN o

o o to to in so in so c^ CM vd- sO so pv rv 00 CO OO 00 CN CN OS CN CN CN CN CN CN CN CN CN CN CN CN CN CN OS CN CN CN CN CN CN O' O' Os CN O* o •— CM sj- in so r>. 00 CO CO CTV O' Os CTv CN CN O' CN CN CN O' CN CN CN CN CN CN CN O' CN CN CN CN CN CN O' CN CN OS O' CN O' CN CN Os CN CN Os O' o

a

V in o so CM 00 sd- O' O' vd- PJ O vd- sO 00 sO in o sd- n- vd- to vd- to in vO pv 00 o vd- in p^ CN O PJ sO CN to sO CN CM vd- O' o

~x> PJ to to *d- sd- so in so 00 r*- 00 sd- 00 PJ in 00 o — PJ PJ to to vd- vd- sd- in in in in in in so so SO SO SO SO p* r-v eo 00 eo CN Os O' O' o

•o o o so to CM o to CM o CM sd- in so n. !>. 00 00 00 CN CN CN CN CN CN CN CN O' CN CN CN CN CN CN CN CN CN CN CN CN O' O' Os CN OS as O' CN CN CN CN o

0) rHCM vd* so 00 OO CTv CT' as c^ CTV CTS CT* O' CN O' CN CN CN CN CN O' O' CN O' Os CN OS CN O' CN O' CN O' CN <N CN CN CN CN CN CN CN CN CN O' O' o

U

CO o in 00 o in CO CO to O m 00 PJ O c^ O in sd- PJ eo PJ in sD pv O PJ sd- sO 00 to h*. CN O PJ sd- in vO p. CN O PJ to vd- in O' o

PJ PJ in sD c^ o c^ 00 r-- O' 00 >d- O vd- 00 o *- PJ to sd- Sd- in in in in sjO so so SO sO r-* r>. rs. 00 00 eo eo 00 00 00 00 CN CN CN os os CN O' CN o

o o PJ O' sd- o r-> to sd- in so OO 00 00 as CN CN CN CN os CN CN CN CN CN CN CN CN CN CN CN CN os CN CN CN O' Os CN O' O' O' CN CN CN O' O' CN OS o

to so 00 CO OS CT' CTS CTV <T» c^ CTS CT* CT* O' CN CN CN CN CN CN CN CN CN CN CN CN O' CN CN CN CN os CN OS O' O' CN CN O' CN CN CN OS O' CN CN O' Os o

in o O' m sD CM in in o o PJ o PJ sd- CM CN O CM vd- in so 00 O to in eo CN CN o PJ to sd- in so CO PJ to vd- in eo 00 o

PJ sd- PJ 00 to so CM sO O PJ to sd- •d* sO so SO so v£> sO r-v n- p. CO 00 00 00 CO 00 00 00 00 eo CN Os CN CN CN os CN Os o

o o so so m sd- m sO so h- CO 00 as CN CN O' CN CN CN CN O' O' CN CN CN CN CN CN CN CN CN CN CN CN CN O' CN O' Os CN CN CN CN CN CN CN Os CN O' o

CM so 00 CTs O' CTs CT* CTv CT* O' CTv CT* O' O' CN CN CN CN CN O' CN CN CN CN CN CN O' CN CN CN CN CN CN CN CN CN OS OS CN O' CN CN CN CN CN OS CN as CN o

o m in CO o to sO vd- sd- h- CO PJ to OO in eo PJ to in SO CN CN o PJ to to vd- in sO SO p. 00 00 O O PJ to *d- in sO 00 eo CN O o

•d- O' in r- CM n. sd- — to vd- m SO sD n. pv r**. r-*. n* pv t^ n- eo eo eo eo 00 eo CO 00 OO CO CO CO 00 CN OS Os CN CN CN CN Os CN CN O' O o

o o O' r-'. to m SO r-*. r- 00 CO 00 O' CT* O' CN <N CN CN CN CN CN CN CN CN O' CN CN CN CN CN CN CN CN OS CN CN O' CN CN Os Os CN OS CN CN Os OS CN CN o o

in 00 O' O' CTS O' c^ CT* CT> CT* c^ C^ CTV CN CN CN CN OS CN CN CN CN CN CN CN CN CN CN CN CN Os CN CN CN CN O' CN CN CN CN CN CN OS CN CN CN CN CN o o

in o sd- o >d- ro CO c^ in 00 in CM C^ in 00 O' O PJ PJ PJ (M to to sd- sd- in SO sO r-v. CO CN O' o O O to vd- in fN. 00 CN CN o o

o »- o o PJ CT* sd* OO CM vd- sO SO CO 00 eo eo 00 00 00 eo eo eo eo eo CO CO CO 00 00 CO 00 00 CN Os CN CN CN CN CN CN CN CN CN O' CN o o

o o CT* sd* sjO h- CO CO O O' c^ O' C^ CTV CN CN CN CN CN CN CN CN CN CN CN CN CN O' CN CN CN O' CN OS Os CN CN Os CN CN O' CN CN CN CN CN O' O' CN o o

00 CTv O' CTs CTS c^ c^ CT' CTV CN Os O' O' CN as O' CN CN CN CN CN CN CN CN CN CN O' CN CN CN CN CN CN O' CN CN OS CN CN CN CN Os OS O' O' CN O' CN o o

ITl o o

Csj PO cr> CO O 00 'd" CT* to K) vtf" >cr Vd" <!• 'd' 'd' lO lT| lO iTl o-j-vTCT'CONTvO'jar^r-. t^oooococooooooooooooooooocooovO\or^«oeocT>avoor- cvrotolo

cOoooOeoooeoooc'tT'CT'O'CT'O'CT'O' «tf-triLnsor^eoeoovoo

cr* ^ ct* cr* cr* o o

o o

C7\ (T* C7** CT* O' O' O' 0» O' O' O' O' O' O' O' O' O' O' O' O' O' O'

O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' CN O' O' O' O' O' O' O' o

O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' o

T5 WlTfOlTjOtOomOlTiO

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lO\Dor^r^ooooc^o^o

000000000 -“
Cumulative

Radial

Energy-Deposition

Distribution

in

Water#

Beam

Energy

=

60

MeV

ITi O O vD to >tf' O so ^ mOs3"Ossd-o»— lOCNJcO"— CSJ lfl O tO OO O' ITl s£> sD CO tO <Ts ITt O' ITl sO — O O -O’ t>- ITl O O •—CO osS-ossa-"— •—cocT'o-- O'loocNJ*— '3‘iniri'— t^toooi^toso o' r^. O' to oo o co o' to o* vo to cT'st-ooocsjtosj-soooo

o o o CM to sd- so CT' CM in CO sj- t" o sd- •— S^- r*. sd- o m o OS to in 00 o to sd- in sD o 00 00 OS OS os OS Os Os O •— CM 00 CM to to to •d- stf* in in so sO 00 eo 00 C7S OS os Os OS os Os Os Os OS Os Os os OS Os OS o

o in ro o lO >d- m o o CM SO CO sd- o CM in CM CO Sj- CM O o CM to 00 CM CM oo •d- «d- sj- CM «d- m to 00 o 'd- d- o K> S£> — SO in to o CM m K) *— in to to 00 so sO so CO sO sO o m in in os O o n* to o SO o CM sr sD 1*^ 00 OS o

000»— •-COtO'd-lAsD CO*— 'd’C^OtOsOO'^C'. — 'Cft^»— si" O sO *— so O to vO eo O CSJ to IT) so O ^* eOCOO'O'O'O'O'O'O'O

•—•—•— cococotototO'j-'j-'j-LriLn so sd eo oo co eo c^' o> o* o' cr* o> o* o* crs o* o* o» ct' o* o> o< o

ITl O

o •-

sa si- m o cT- .- sjf so in CO t'*'jooirit^oo0'0 — '0*—irir-.iri'3- o to sa CM r*- sS* o O' o O' cjs o to to >o scr •— o o« m •— oo •- "ct" o

o o so to — vo CO m O' o to o CO o •- t-s lo so m •— •— co m cm o

cot^r^csjo*— •— ocot^oocom*— sa co O' <si ss- so n* co o'

o

•— •— o o o CM to to m in r%otO'aotor'-»->ct-eotMmc7'(M'a

r- — — — <MCMtOtOtO'J-'3-'3-inin

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•— to sj- in sa t'- cocoo'<t'ct'0'C7>0'CT>o ^'at'^n-oocoooo'O'O'O'O'O'O'O' O'O'O'O'O'O'CT'O'CT'o o m

o o

C7' va in to va m to so co •— to o' in eo c^" in o to to in sa to ct' oo o CM sa *— r'"i- to to to sa vcc ns t*. in in to O' c?' •— m •— in r*. n*iO'ah-.CMin'3-ocMCT"aocT'r'v*-

'OCMCM'3-o'ooinn"j-in'atocotoeo

inr^cM'3-'d-cO'acM'ao

inct'toin'ar^cocT'CT'o

CM to -cf in so CT* CM in CO CM so o sd" •— >a c7' to sa o

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o O o CM SO so h- sd* o n' 00 n- sd* o sd* r- CTs CM to sd- in in so sO sjd h>. 00 00 00 00 «o 00 00 00 CTs C5S c^ os Os OS OS Os CT' CTs Os o o

1 — — CM CM to sd* sd- m so n. 00 00 00 00 as CTs CTs CTs Os CTS Os CTS CTs CTS CTs CTS OS OS c^ Os OS OS CTS Os C^ Os Os Os CTS Os OS Os as Os OS o o

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1 o o — sJ- O so sd- CM Os so to to SO Os to sd* in in vO so n- r-. 00 oo 00 CO OO 00 00 00 CTs CTs as OS CTS os os Os OS Os c^ OS Os o o o

— — CM to sd" sd* m sO r-s 00 CO 00 CTS os CTS Os CTS CTS as CTs Os c^ Os Os CTS Os OS CTS OS CTS CTS CTs OS CTS as CTs CTs OS O' as Os OS OS OS Os o o o

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Cu in o r— *— CM to f— >3- n- «3- sd* CM so m sd* — — sd- in 00 ro 00 sd* CTS CM to CO in SO m in O SO c^ to in o in 00 CM V— r— n- nw O to sd- Os o o

D CO to sO in so in to CTs CM 00 CTS 00 sd- o in sd* o in o to in 00 Os CM sS- so eo CTS o — CM CM CM to to •3- sd* sd* in so r- h- 00 CTS Os Os Os o o

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00 r-v 00 Os o CTs 00 sd- CTs nD o to sd* in vO so hs. n- 00 00 00 00 00 00 00 CTS CTS O' CTS CTs CTs CTS OS os c^ OS Os Os CTs os CTs Os Os Os o o

CM to in in NO r>. r-'. 00 Os CTS CTS os as CTS Os Os CTS CTs Os CJS CTs Os CTs CTS CTS CTS CTs OS CTS OS OS Os Os CTS Os CTS Os CTs ON OS Os OS Os Os Os o o

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CM CM CM c^ CTS O CM NO CM m Os CTS sj- in — in 00 •- CM sd* ir> so n- 00 00 CTs OS •- — CM to ro sd* in m so I"* 00 00 Os OS ON OS CTS OS Os o o

o o d- in 00 CM CM o in 00 to in so n- C'. 00 00 00 00 CO 00 00 CO 00 00 CTS CTs OS CTs c^ os CTS CTs CTs c^ CTS c^ OS CTs c^ Os Os OS CTS Os Os Os o o

— to sd* so 00 00 00 as OS CTS CTS CTS CTS CTS CTs CTS CTS CTS CTs os CTs OS Os OS CTS CTS OS as CTs CTS CTS as CTs Os OS CTs Os OS CTS OS OS OS C^ Os Os Os o o

in o CTs CTs CTS oo r— r— to d* sd* CO n- SO O CM CTs O sO f-N sT IT- CM 00 CM sO to O d- CO O to CTs d- C^ O CM CM CM to >3- <7s CM to in Os o o

— CM SO CTs in o 00 NO d* in to 00 •- to in NO CO as •- CM CM to to 3- in in m vO sO sO n- n- 00 00 00 CO 00 CO 00 OS CTS CTs CTs Os Os o o

o O O to to to 00 to d- in so n- 00 00 00 eo 00 CO CTS ON CTS CTS Os ON ON CTS ON C3S C^ C^ CTS CTs CTs Os Os CTS CTs as CTs c^ CTs Os C^ Os C^ c^ Os o o

to m CO CO Os CTS Os CTS c^ CTS Os CTS OS CTs ON CTs CTs CTs CTs CTs CTs Os CTS O' os Os CTs Os CTS OS CT' CTs c^ CTS C^ Os Os Os Os Os Os CTs Os CTS OS Os o o

o m sd* to to 00 so in O in o 00 sO o SO OS CTs SO CM in n* CTs v3- sd* SO o CM CM sd* d- in SO 00 Os o to in n- eo to CO O to OS O o o o

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o o o 00 in sd- in so NO n«. 00 00 CO CO CT» CTS c^ CTS Os ON CTS CTS CTs CTS CTS CTS CTs CTS Os CTS Os C^ as C^ OS c^ c^ CTs Os CTS CTS O' Os CTS CTs Os o o o o

to NO 00 CTS c^ CTs CTS CTS CN CTS CTS OS CTS c^ CTS CTS CTS Os CTs CTs C^ Os CTS C^ Os CTs as OS Os CTS CTS OS c^ €^ Os c^ os CTS Os c^ OS Os CTS CTS Os C^ o o o o

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Radial

Energy-Deposition

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in

Water,

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in ro Os o ro CM sd- sD sd- in to SD O o SD crs to sD o id- id- -d-ineoso.— sd-sosoto m to o <j- m tO'd-ineohiococy'CMo o SD ro cr^ m Crs 'd* to m os o SD — — t*, <f sD sd- vD Os tn sd- crs os cosd-i'^in.-son.coto to >d- sd- <- o in id-.— inoOsOhiCOCMhiO

o O r- r- CM to sd- in *D CO ro in r- vd" •o sd- hv o sd- sD O sOCMsO — socootoin sD n- 00 O •— r- toinsosohicooocy'cy'o

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r- o o o o r~ CM to >d* in sD 00 o to SD CO in CO V— sd- ns o sd- ns »— com*— morosDcrso CM to in in so so nsn-oococrsoscrsoscrsc

•— •— CM CM CM to to to sd- sd- sd- in in^t'ir-FOOcocococy' CTs Cts Ds D> ^ososcrsososcrscrscjso

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•— r— CM CM CM ro to sd- id- id- insoson»nFeococy'<J' Os Ds CTs Ds CTs Cts OS OS OS cr* crs os os crs os o

O K> sd- <1- c^ n. 00 sD CM CO CO 00 OS OS CO CO in CM O CO incFFid-eosoor-oCM so t"- n- o inn*ossd-rosDOCMsDo

o O f— CO in 00 rO 00 ro CO in ro CM O 00 O o> CO o sD CM sD CO to .d-CMr-Fso.— oocMco to Cr> CTs CO -d- loinsohicococy'cyicy'o

O o o o O CM to ro in hv OS vd- sD O CM in c^ CM sD Os CM so CMosin.— i>-cMsocts.— <1- in so [S' eo CO Os Os crs Os OS Os 0> Os CTs O

r- CM CM CM CM ro to ro Sd- sd- ininsoi^n-oocococy' CTs crs Q\ cr* Ds os Os Os Os Os OS OS Os os Os O

m sD 00 CO SD in CM 00 os m Nd- sD to OS in CM o in o o to in in ,-CMCM^sd-CMCMr-sd- co cys so Os ^ so hiCM—•COCM'd-t'^COCT'O

o> O *- CM m 00 CM to Os sD — Os O 'd- n- CM O sD OS CM in n* crs ooor'-lOlncMtooc^ o eo CM CO CTsvOCOCO^'^<^<7sOso

o o o o o O CO CO ro in sD C7S to sD crs «— sd- ns o ro r-t>-tOCTSid-0'tOr-«CJ' CM to in so r" COOsOsOSOsOsOSOSOso

•— T“ CM CM CM ro ro ro Id- id- ininsovof'-i^cococo CTs Os ^s crs OS OS Os CTs Os CTs Os CTs CTs O* O' O

o SD 00 OS <r* O o CO o in CO O CM in in 00 n- in CM to eo o hi CO sOIOOt-F-COtOOCM.— O' CM CO O CO co'ocy'hicy'id-hieocy'o O ro vD o ro n. to 00 ^d‘ sd* sD CM CO •d- to to m O' *- — ns 00 O id-soiosoosotomio ns o crs sD 00 vD oosd-n^oocooscrscrscrso o o o o o f— CM CO to ^d* sD 00 to \0 CJs CM in 00 CM in CO in r-f^toooioi'i.— sd-h~ O' CM to m vo hi co^crscjscrsososcrsoso

•— ^ «— CM CM CM ro ro ro sd* sd- ininsosor^e-F-eococo 00 CTs OS OS crs crs osososcrsososcrscrscrso

m sD Os 00 c^ m 'd- 1^ to CM in O SO id" 'i- to CM r-s in 00 Sj- to toeosd-vo.— r-.-in.— in CM Ch CM to inioineocijiiOh-t^coo

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CO o sj* 00 CO n- o CO sD sD CM o >- .d- vD — vD 00 — m sd- vD sOtOin.— lOCMsDCMsO 00 sD ro 00 o o sDtOvDn^OOOsCrsOsCrso- o o o o o t— CO to to si- sD CO sd- o to sD O to sD o ro sD CTs ini— sDir-mcrscMinns Os s— ro sd- so n- co^^osososcrsoscrso

•— r— fl— CM CJ CM to to ro sd- sd* sd- sd- insosonFr-Fi-Fcococo oo crs CTS cs crs crs CTs CTs Os Os crs O' O' O' O' O

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Cumulative

Radial

Energy-Deposition

Distribution

in

Water»

Beam

Energy

=

15

MeV

in o — 00 in in 'd' to CM in so ro o ro o •d“ ro M3 d* CO •— w~~ d* O' ro in ro tn d- CM d- to d* O O 00 •43 rO fO 00 o r-v 00 CT o o

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in m o in in O' \0 m sD CM oo M3 CO CO to CM 00 in CO M3 CM so SO o 00 CT in 00 O' CT n- d” *“ CM 0- m ro CT d" n- 00 CT CT CT o o

o o o o •— CM ro m C^ r— S3- O' in — SO CM to 00 CM so CT' CM m CT ro in CT o CM to d* in in •43 so oo 00 CT CT CT CT O' CT o o

— — •” CM to ro •d" d- in in sO sO sO n- n. hv CO 00 00 00 CT CT CT CT CT CT CT CT O' CT cs CT CT CT CT CT CT CT CT o o

in o CO 00 sjD to •sO o 'O O ro in so 'd” 00 ro CM 00 CT' to O' r>- t-^ CM nv CM O to to (M CM so CT O sd* so d- n- sD 00 sd- 00 CT O o o

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o o o o CM ro in o ro M3 O 'd- 00 'd- o m CT' CM sO 00 o CM <1- in 00 o CM ro ro d* in in so so r-. 00 CT CT CT O' CT CT O o o

CM CM ro MT sd* in M> •43 SO n- r>. 00 00 00 00 00 CT CT CT CT CT CT CT CT CT CT O' CT CT CT CT CT O' CT CT CT CT O o o

o in m o o CM in CM 00 M3 in •d- SO CM O O' o 00 o SD CT d“ (M CM O in o CM d" in d- ro ro SO CM CM o sd- 00 CT o o

sd" ro ro c^ CM O' CM CN CM in CT' 00 00 CM CO — O' CM in CM in to CT' *- CM CM so so in to oo CM so CT CM d* so 00 o CM •d" 00 CM SO 00 CT CT CT CT o o

o o o CM in n- >3- c^ to rs. M3 in 'd- o — SO CTS CM d- so CT' o CM to sd" m in SO SO SO n. 00 00 00 00 CT CT cs CS CT CT CT o o

•— CM CM ro 'd* in M3 so 00 CO CO 00 CO O' O' CT CT O' CT CT CT CT CT CT CT CT CT CT CT CT O' CS CT CT CT CT CT CT o o

in o m in CM sd" O O' •>3“ 00 O' in in CM O' O' to t-s in CM O CT CT to m o in eo ro CM CM so in in d- 00 CT in «3- SO sO CT O o o

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o o O CM Sj* 00 CM to 00 vJ* O' o O' ro CO CM d* CT' o CM EO ro ro d- in in so so so h*. Tv 00 00 00 00 cs CT CT O' CT O' CT o o o

— CM CM ro ro in in so n- 1*^ CO CO CO CO O' CT' CT* CT* CT CT CT CT CT CT CT O' CT CT CT CT CT CT CT CT CT O' CT CT O' CT CT O' o o o

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o o to CO o n- in CM c^ in M> in o m CO O CM ro d- d- in m sO SO so n- r-- 00 00 00 00 CO 00 00 CT CT cs CT CT O' CT CT CT CT CT o o

CM CM ro 'cr 'cr in SD hv 00 00 CO CT' CTs O' <Ts C^ CT' O' O' OS CT O' CT CT CT CT CT CT CT os CT CT CT O' CT CT CT CT CT CT CS CT CT CT o o

o iTl o ro O \D M3 O' CO CM O' so CT' 00 so K) o m m O d- m ro CM d" CT CT in SJ- in to o ro O 00 sD SO CM CT CM sd" oo o o

CM ro to CM o O' — CM n- CO to in CO — O' d* o m O' - rO m r-- eo o CM sd- in so 00 CT CT o •- CM to d* sd" m in •43 00 00 CT os CT o o

o o sO 'J- ro to 00 ro o CM d* d* in sO sO so hv 00 eo 00 00 00 00 oo 00 CT CT O' O' CT CT O' CT CT CT CT CT CT OS CT o o

CM ro in sO M3 00 00 CT' C^ CT' c^ O' O' O' O' O' CTS O' CT* CT CT CT CT CT CT CT CT CT CT CT CT (T CT CT cs CT CT CT CT CT CT CT CT o o

o iTl 00 in O' in 00 00 CM sd* ro CO 00 •d- CO d* 'O to so O' to CM CO CT CM CM CM CT sD SO CM SO CT ro 00 sd" sd" o in CM sJ3 O o o

CM \D ro o o O' *- to m o r- 00 sD so CM •M3 o to in oo <T' *- CM ro in sO 00 O' O O CM CM to to •d" sd" d" tn tn sO 00 00 CT CT O o o

o o to ro r-- CM 'cr* sO o in 00 •— ro 'd- m so SO 1^ 00 CO eo CO 00 00 CO CT CT CT CT os CT CT CT CT CT cs CT CT CT O' cs CT CT O o o

CM 'J' in \D 00 «0 00 CJs O' C7N CT' CN C^ CT' CTS O' O' CTS CT* CT* CT CT CT os CT CT CT CT CT <T CT CT CT CT O' CT CT CT CT CT O' CT CT O' O o o

in o vD in o c^ %£) O CM in to CTv o CO O' to in 00 d- 00 00 h> CM in O CM CT m CO SO CM 00 •O CM sO CT O CM O' CT so 00 CM •43 s£> T- o

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o o 00 O' c^ m OS CO to sd* in •43 rs CO 00 CO 00 00 00 CO 00 eo CO 00 CT CT CT CT CT CT CT CT CT CT CT CT CT O' OS O' CT CS CT O' CT CT o

CM m \D r-. 00 00 O' O' O' O' C7' CT' O' O' c^ OS cs O' o O' CT* os CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT CT o

o m O' o 00 o CM o 00 O' 'd' CM in in d- CM CO to SO CM ro rO in O CM in CT CM SO to CT d* CM sd" 00 O o o

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o o so ro CM O CM in m sD M3 Cs 00 00 00 00 00 00 O' O' CT' CTS CT' CT CT CT CT CT CT O' CT CT CT CT CT CT CT OS CT OS CT CT CT O' O' O' CT o o o

CM so 00 00 O' os O' O' O' O' os O' CT' CTs O' O' O' cs O' O' O' CT CT CT CT CT CT CT CT <T CT CT CT O* CT CT CT CT CT CT O' CS CT O' CT o o o

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Cumulative

Radial

Energy-Deposition

Distribution

in

Water*

Beam

Energy

=

15

MeV

in o so to sj- O' o (O to os O' to eo sJ" o' so s^- os sd- so so m to csi cm •— eo cv to os eo rs- sS" eo o' m o* to to to >d- •— so <o o

«— CM CM to to so CO os o so — m r- m os o to o sj- CO CM so •— crs s3

- eo in cm o cts o o ss- os crs cr. o so o eo C'- sd- eo •— to so o ©

T— o o o CM CM to to N0 CO o NO c^ CM in r-v o 9-* in o o sd* ON o to in NO CTN ro vO 00 to 00 ON CMsor^*— inson-oocj'©

r~ CM CM CM to to to to Nd- Nd* sd- sd- Nd* in in in in in nD *<£> s^ nw r*«. r-^ COCOOOO^CTNCTNaNONCTNO

O in CM in CO o o NO CM CM in CM CO CO CO CM NO CO NO ro ro in CTN NO CO 00 vO o Nd* <- 00 CM lO CM t'«eo©roosoe0'— CM©

o 00 CM O to CM in o CO to vO NO 00 to NO o in to NO CO o NO to s3- 00 s3- to in in to 00 SO Nd* r-H c7'CMt-^©oOLnoooo*-©

r- o o ,1- CM to <r in in n- o to in CO CM to in CO to NO ON CM in r«. o n. o to SO CTN m CTn o *- CM to in NO r- Os«— tOsOvOI''.f^C0CJ'©

r— CM CM CM CM to to to Nd- Nd- Nd- in in NO NO sO NO n- 00 00 00 00 00 00 CO COCTnOnCTnCTnONOnOncTnO m o CO ro nJ- c^ nJ* o C7N o !>. CTN c^ 00 o n- CM o r-v n. to m o NO to NO ro CO SO Nd- r-^ to to CO o so CM lO o CM oototoeO'3-s3-CMsor'^© O o to sD to CTn o ro Cn. CTN to o CM Nd- n- NO o in n* c^ — m ro On o 00 O CTn to 00 o sd* in CM hv m CO CM cT>*d*osd*r''-tONd-Lncoo

r- o o o r- to Nd“ nT in in CO to NO cr» to o to c— •d* n- r- to o sO CM 00 NO 00 CM to sd- Nd- in in NO NOr^COOOCOCJSCTNCTNCTNO

CM CM to to to Nd- Nd- Nd- in in NO nO n. 00 OO 00 ON CTN os CTs CTN CTN CTN CTN On On On On O' CTN On OS O' O

o in CO O in NO oo ON On ro n. 00 CM o Nd“ r-v o ON NO ro tn o n. ro CO 00 CM O ro O sd- NO NO O omt'vs3*CMI'»©>3-©©

o o o CO sD to CO NO to CO CO CM in CM CM CM in cr* ON •- CO o ON Nd* to in NO NO ON Nd* Ncr O sd- in 00 to vO CTN CMrO'S-sOI-^I-'-eOCr'©©

«» o o O CM CM to sr m NO CO CM Nd- o to NO ON ro NO CN Nd* NO o sd- V— CM NO o CM Nd* in NO n* 00 00 00 OnOnCTnONOnOnCTnOnoO

*—

•

«r— CM CM CM CM to to to Nd* sd- m in nO n. CO 00 c^ ON ON ON CTN c^ CTn CTN ON OnO'OnOnO'ONONOsoO

m o in o o to NO nO CM CM NO CO CM to Nd- in in ro o ro c^ to oo to to NO sd- 00 n- sd- o r*^ s3- sd- to '3-eoi'-ro©«— CMinvoo

ON o o sj* rNs o in ON '^r to CM o r>. On >d- to to o CO — in On Nd" CTN o CM o — ON CM in vO to sd- — tn c^ ss-inr-^ooo'^tJ'soscT'© o o o O o r-. r— CM to Nd- in Cb sd- o to in CO ro in CO CM in to o NO r-v r- vO CTN CM s3* vO i-N. 00 00 oo On on On On CTn on On on On O

r— CM CM CM CM to to to sd“ Nd* in so nD r>- 00 CO 00 CTn CTN CTN CTn CTN <7N CTN ^ O' O' O' O' O' O' O' O' o

o in K) CM CM NO NO to ro in m <Tn in 00 On Nd- NO CM c^ in to o o to ro ro CM in NO CM sd- rs. m NO <3- n. CTN NO eot''©so©CMs3-sor-'0

ON ON o •- K) m 00 Ntf* 00 o Nd- — 00 CM n* m to — CM NO ON CM Nd" SO — to to to Nd* m o c^ — in in CTS 00 NO ro (Mn00000<^CTNCTsCTnCTNO

O o o o o o o T~ CM to sd- NO o CM in CO *— Nd- r-. o to hs. o to o vO CM 00 to 00 to NO O CM s3- tn vO 00 O' O' CTn On On CTn CTn On O' O

«— «r— CM CM ro to ro to sd* Nd* in m sD i-N. CO 00 CTn ON CTN ON ON CTN CTN On On O' O' 0^ CTN O' O' CTn

'in o m On 00 sD Nd" SO CO CM 00 o in 00 CM Nd* o to NO o to so to NO to CO Nd* o 00 00 CN O tn NO CMs3-^'^lnn.l^el0<J'C^©

CO On o r- ro O ON CM m o tn o in CO 00 in *- o CO CO NO CO — *- sd- in in NO o to t-s CM o — •- ON in NO in OnOOO^^ONOnOnOno

o o o o O O o CM CM to nJ- SO On CM in CO o to NO ON to NO On CM 00 S3- o NO in o Nd* CJs CM to m nO t-N, CTvOnCTNOnOnCTNCTnCTnCTnO

•— Cd CM CM ro to to to <t* sd“ in so 00 00 00 00 CJS On c^ c^ c^ ON O' O' O' O* C^ CTN O' On

o in K) CTN <r c^ in CM rs CO o t" to o Nd- o CO m SO m 00 CM n. CM to tn CM r*v CTN s3- incMtotoinr^i^cT*oo

00 00 o CM in CO CM CM 00 Nd* 00 in to CO to — OS NO NO NO n. o — 00 Nd- CTN o in NO to sd* *- in NO Nd* CO -- eomoocT'O'CTscj'a'©©

o o O o O o o r— CM CM to sd- NO 00 o CM in CO o to NO ON CM in CO CM CO to OS *d- o Nd* CO CM tn oo o CM Nd* in (OCTNCTnONOnONCTNCTNOO

•— «— **“ CM CM CM ro to to to 'd* Nd- in m NO t-' 00 00 CO CTS On CTn c^ On CTn O' O' O* O' O' O' O O

in o ro m CM to to CO Nd' NO CM CM NO CM Nd* ro Nd- ON m rsi CM CM CO o sd* to 00 in r-^ S3- O CM mvoooo''3-soeooocFso

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r— •— CM CM CM CM to to to Nd* sd- in in NO so r*^ 00 00 CO C^ CTn CTn CTN ON O' O' O' 0^ O' O' CTN O' O'

o in c^ sd* CO to m to to ON 00 c^ a> m in o in to o O' in tn 00 NO NO o sd* sd- nO sd* scT o r*** 00 o CM 00 in to ON •— 00 NO 9— NO SO 00 o

r*^ o CM sd* CO CM r*^ to o r*^ in in in — 00 ON CM d* NO «o CTn o •- d* r>. to CM r>. CTN 00 o 00 to m d- — in 00 O' ON o NO n- oo CTN c^ CTN On o

o o o o O o r- •— CM to ro sd* NO 00 to NO O to NO O' CM tn CTN CM in 00 Nd* o m CTS to r-- 9— to so 00 O CM to Nd* m CTN CTN ON c^ CTn CTN CTN CTN o

9— CM CM CM CM to to to sd* sd* sd- m NO «o so r-% 00 CO 00 00 O' CTN O' CTN OS O' ON CTN CTN CTN ON CTN ON OS o

m o in CO in ON CM d- to CM ro 00 NO CM O NO 00 O' so o to sd- NO d* o d* CM NO to NO d* Nd* in Nd- Nd* Nd* NO CM in r-w O o

o CM tn CTn m f— CTN ex' to CM CM NO 9— to o tn o in o O ** o o d- in o CTn t-- o CJN d* O' to to to CTN CTN in 00 c^ CTn c^ O o

o o oooo*— CMCMtOsj-in oo»— 'd'n-s— sd'co«— meocMinoscjin •— soosd'oootosoooo'*— CM'S'inso n-coo'O'O'O'O'O'oo — CM CM CM to to to sd" 'd‘ sd" in m so so n- oo oo oo co co os cjs o> cjs os O'O'O'O'O'O'O'O'OO

o in

so so co«-ini''CO<ysocMinsd- otorsCMoscT'ocMsd'C's

CM — — CM in 00 1'' so O' m CO os to cm m cm o crs co to .- © sd- m cm sd- cm oscjcy'©oinsosot''0'>d'ocMsd‘CM >d’n''OO'C0CMto*— t'»©«— •— «— or^

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CM CM to to to sd" sd" in in in so so so co co co co css os crs cts O' crs OOCOCT'CT'O'OSCTSO'CTSO

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Cumulative

Radial

Energy-Deposition

Distribution

in

Water.

Beam

Energy

=

10

MeV

in o 00 stf" in V— h> in CSl 00 to o so 00 O' «vi- O sO 'd' in m sd- C7S hv in vD in fs- CM m in o o 00 sO in O' O' O O

in sD o ro r>. CSJ O' sr> m SO O' SO so — so in in CM — sO *- SO 00 n- to to 00 00 in 00 O' 00 sj- O' to so O' O' o to CM sc:; CO O' CTS o O o

o O o o o CSl rO in VO O' ro so CTs to in 00 (SI in 00 n- CM n. sd" o CM sd- •o 00 O' (SI to sJ* n* 00 CTs O' CT' <T' O' CTv O o

(SI CM to to to s3* sd~ <r in in in so so n- 00 CO oO 00 00 00 (7s O' c^ CT' O' O' CT' O' CT' O' CTs CTs o o

o in CO oo m ro sO (SI «o to to o in sd- o ro CM in O' 00 to to 00 n- (SI ro O in n- 'd- O' sD 'd' <r sO 'd- 00 CT. o

in in o >3“ o r>. o si- 00 to CM >a- eo (S) SO c^ 00 n>. VO to 00 SO S3- CT' n- CO sd* n- to o^ in 00 C7S o O' eo n- to sd" CM sD 00 O' CTS CTS C^ o

oo oo^r-cJrOLr^vD^^o^ inosi'dOCsjvDO'cJ-r^otOsD

•— r— CNjc\itOK>^o«cTv3“tOLnLn^D'.cs£> t^t^r^oooocooooooo^^o^o^^o^ <yscr*c7^o^c^ONC7scrvaNo

in o o sd- CM 'd* to o SO vO to n- CT' as o 00 ON n. o CTs sd* CO sd* r*^ CM CTS o CM sd* to m sd* OS 00 O' in to V— Tv *— ro o o o

'd' m in (SI to o O CM so sD O CM c^ o 00 in 00 m o o Os o to 00 sO sO so sd* in n in sd* t'' to CTs sd* so in in 00 CT CT o o o

o o o o CM to in CTs to 00 sd* CTS vl- o in CTs sd* 00 CM so CTS so O' CSl sd* so OO ON o CM to sd* sd- in in so rv. 00 CTs O' CT <T CT o o o

*” (M CM ro sd* vl- sd* in m sD SO SO n* n. n- 00 00 00 00 00 c^ os CTS CTS CTS CTs O' os CTS CT CT CT O' CT CT CT o o o

o in o o so to vO CO Cr> to CO sD to 00 <T as in 00 00 CM to sd" SO as 00 so sd* t-v CTS CTS 00 in sd* >i- in in in to o CTs to in o »d* SO 00 o o o

-d* sd“ (SI 00 00 'd' in 00 'd“ to (SI sd* o m O n. sd* to in CM CM as to (SI 00 to IT{ to o — o 00 in so - vO so O r*H CSl so 00 CT ON o o o

o o o o «-> to in o to so CTS O ro CTS sn hs (SI sO o to in 00 o to in n- 00 o •— CM (SI to sd* sd* in in so so 00 00 CT O' CT CT CT o o o

r- r— f— (M to to sd* in in s^ sO n- n- n- n- 00 00 00 oo 00 00 as O' CT* CTS CT* CTS ON CTS CTs CTS CT CTv CT Os OS CT CT CT O' o o o

in o

to vd* rocMsd“inso*-hvscr(MfMtofOCMtotor^sootoo

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o o eo (SI sd* CT (SI ro *d* in in sO so Tv Tv 00 00 00 00 00 00 00 00 00 00 00 00 00 eo CT CT CT CT CT CT C^ CT CT CT CT OS CT CT CT ON CT CT CT CT o

in Tv eo 00 CT CT CT O' CT as CT CT CT CT CT Os CT CT CT as CT CT CT CT <T CT CT ON CT CT CT CT CT CT CT CT CTs Os ON OS CT CT CT CT Os CT ON CT o

in o 00 ro o o in CT (SJ in 00 o in o Tv so eo CM so 00 sd* CT CT sd" sO 00 (SI sd* tv m sd* Os CM in rv CT CO sd* CT rv o O O O o

o to s£> sd* sd* o in o to in o CM sd* in so (O CT OS CT o o o o -- (SI ro CM CM to to sd* 'd* sd* in in in in SO rv eo 00 CT o o o O o

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si> 00 CT CT CT CT CT CT CT ON CT CT CT CT CT CT CT CT ON CT ON CT CT CT On ON ON CT ON CT CT Os CT CT CT ON ON ON ON CT CT CT CT Os CT o o o o o

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Cumulative

Radial

Energy-Deposition

Distribution

in

Water,

Beam

Energy

=

10

MeV

in o o stf- CA CM o Crs r^. o CM sl- in vd- sl- CM CO o c^ sl eo in O O CM CO sl O OO CM n* to CM o o — CM o o o o o O o o o o o O o in 'Cl- vO to c^ sl* so CO CM sl* to to to 00 h- sl- sl- sl O to o to eo O o CTS sl CO in CTS in vd- n. CT' tn o

r* o o o o o o o o o o o O CM n- o vd- so CO o CM to 00 in o •— m t'' CTs o to so r-^ CM r-> O' o in 00 CM 1- SO CT' o •— »— CM CM CM CM CM to sl- -cr Sl* sl- sl in in in in so SO sO r-- t-^ n- 00 eo 00 CTS O' cr> c^ CTs o

o in CM to CM o Os CM CO so si* os CN to CN vd- o OS in so OS CM CM to so c^ sl tn in CM sl so O' to 00 in CTs to o

o o o o o- to sT sO so CM OS r>» sj* o O to Os eo to sO d- O' C7» lO CTs to o in sl sl eo CM 00 00 eo CTs o eo CM eo o CM >1 so -1 o

o o o o o CM CM CM CM CM d- in to >d- o f— in sO CO O '1- OS SO CO O' to sl- eo r- r- CM o CM r-v OS to sl t-^ c^ sl in 00 00 O' o •*“ CM CM CM CM CM to to to to *1- sl- sl- in in in SO SO SO sO r>. eo eo CO eo O' CTS Os CT' o

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Cumulative

Radial

Energy-Deposition

Distribution

in

Water,

Beam

Energy

=

5

MeV

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Cumulative

Radial

Energy-Deposition

Distribution

in

Water,

Beam

Energy

=

4

MeV

in o ro ro hv 00 'cJ' iTj m m oo o oo co ro •j- m o so o o oo o on m fo to c\j r>. oo o ^ ir> k) o oo ^ m cm oo co o o o

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o o o o o o CM to 'Cf sDt^O<Ms^T^OtOinoO»-'d-KOCM

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U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

SHEET (See instructions) , 82-2451 January 1982

4. TITLE AND SUBTITLE

TABLES OF ENERGY-DEPOSITION DISTRIBUTIONS IN WATER PHANTOMS IRRADIATED

BY POINT-MONODIRECTIONAL ELECTRON BEAMS WITH ENERGIES FROM 1 TO 60 MeV,

AND APPLICATIONS TO BROAD BEAMS

5. AUTHOR(S)

M. J. Berger and S. M. Seltzer

6. PERFORMING ORGANIZATION (If joint or other than MBS, see instructions) 7. Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP)

10.

SUPPLEMENTARY NOTES

Q21 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

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

bibliography or literature survey, mention it here) This report presents tables of elementary three-dimensional absorbeddose distributions in a water phantom irradiated by monoenergetic

,

point-monodirectional electron beams. Such distributions have been

obtained by the Monte Carlo method for 14 beam energies from 1 MeV

to 60 MeV. The tabulated results can be applied to the determination

of absorbed-dose distributions from parallel beams of arbitrary finite

cross section. The beam of interest is treated as a superposition of

point-monodirectional beams, and the absorbed-dose distribution is

obtained as a corresponding superposition of elementary absorbed-dose

distributions. By way of example, the tabulated data are used to

obtain (1) depth-dose curves and practical ranges in broad-beam

geometry, and (2) central-axis depth-dose curves, radial distributions

of absorbed dose, and isodose patterns for beams with finite circular

cross section.

12.

KEY WORDS fS/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Dosimetry; electrons; Monte Carlo; point-monodirectional beams; superposition;

treatment planning.

13. AVAILABILITY

[^)g Unlimited

I I For Official Distribution,

14. NO. OF

PRINTED PAGES

Do Not Release to NTIS 54 55

Order From Superintendent of Documents, U.S, Government Printing Office, Washington, D.C.

20402. 15. Price

Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $ 9.00

USCOMM-DC 6043-P80


1I\I