Stopping powers and ranges for heavy ions

Takeshi Hiraoka and Hans Bichsel

National Institute of Radiological Sciences, 9 — 1, Anagawa — 4, Inage — ku, Chiba — shi,

263 Japan *22nd Avenue East, Seattle, WA98112 — 3534, U.S.A.

Reserarch Code : 201

Key words : Stopping power, Range, Heavy ion, Mean excitation energy, Energy loss

Abstract

Stopping power and range functions have been calculated for carbon ions and are presented in the

form of tables. With a suitable interpolation scheme, intermediate values can be obtained. The

uncertainty of the values is of the order of 2% at high ion speeds, but much larger at small speeds.

1. Introduction

With the initiation of radiation therapy with 290 MeV/u carbon ions at the HIMAC (Heavy

Ion Medical Accelerator in Chiba) in June of 1994, the need for tables of stopping power and range

for these ions has arisen. Stopping power is needed for dosimetry with ionization chambers and for

the calculation of depth dose functions. An extensive set of tables for all ions from He to atomic

number equal to 103 has been published by Hubert et al.1) for many elements, but no recent tables

for compounds are available. Earlier calculations were published as UCRL reports at the University

of California in Berkeley in the 1960s2).

For the use in cancer therapy, data are needed for compounds. Therefore, calculations have

been made with the algorithm described in Section 2, and Tables are given in Section 3.

2. Method

2.1. Theory

No fundamental theory of stopping power S for small particle speeds (for ions with charge

number z < 12, (3= v/c < 0.1, corresponding to particle energies T < 10 MeV/u) is available at

present. Sigmund3) described this problem, and proposed an approach which is still under develop-

ment. The algorithm used here for these speeds is that of a function with several parameters which

are adjusted to approximate experimentally measured data°. The fact that this function is described

as a correction term to the Bethe theory is only a convenience used to reduce the dependence of

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the correction on particle speed. Other functions such as that proposed by Brice could serve as

well. Evidently, further accurate experimental measurements of stopping power are needed,

especially for gases at energies of a few MeV/u.

It is important to realize that the residual range of a 10 MeV/u carbon ion is less than 0.5 mm

in water, and thus is only a small fraction of the total range of 160 mm of a 290 MeV/u ion.

For the biological effects the energy deposition by these ions will be important, but in an extended

Bragg peak only a few of them will appear at any one slice of the absorber.

As the starting point for an approximation of S, the Bethe theory is used here. It is at present the

most reliable theory of stopping power, derived by Bethe5).

where,

v = 3c = the speed of the incident particle,

y2= 1/(1 — p2),y = 1+ T/Moc2,

ze the charge of the incident particle,

e the electron charge,

m the electron rest mass,

No = 6.022045x1023= Avogadro's number,

Z the atomic number of the absorber,

A the atomic weight of the absorber in g,

B the stopping number and

Mo the rest mass of the particle.

Either 3 or T will be used as the variable indicating the particle energy.

Bethe was able to derive an approximation for the stopping number B in the form

Bo (3) = ln 2mc2132y2 /I — (32

where I is the mean excitation energy of the absorber, defined by

with f (E'0) dE = 1, where f(E,0) is the dipole oscillator strength, DOS (which is related to the

0 optical absorption coefficient).

As far as the incident particle is concerned, this expression depends only on its charge ze and

its speed v = f3 c. Thus, for large speed, the expression should be valid for all ions. In this deriva-

tion of Bo, Bethe used sum rules which greatly simplified the calculation, but which resulted in an

approximation for small speeds. Later he used a different method, finding a need for what now is

called "shell corrections" 6). Further corrections were derived by Bloch') and Barkas et al.8). A full

discussion of these and other corrections is given in Ahlen9) and Bichsell()).

At small speeds, positive ions will pick up and lose electrons, thus their charge is reduced. A

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recent study of this effect has been made by Hubert et al."). Their expression for the factor x giving

the reduction of the charge of the ion is based of empirical study giving a set of parameters used to

calculate x. For present purposes, this parameterization for x is used.

2.2. Materials

The materials for which data are given have been chosen to be useful for applications in radia-

tion therapy. In particular, absorbers may be used outside a patient to inform the radiation field to

the desired shape covering a tumor. Also, absorbed dose is frequently determined with ionization

chambers. Furthermore, a simulation of patient geometries can be made with tissue substitute

materials. Absorbers frequently used for this purpose are listed in Tablel, and stopping powers and

ranges for them are given in the Table 2. All densities are estimated for 20 degree centigrade in

g/cm3.

2.3. Accuracy

No direct comparison with experimental data has been made so far for this report . The

parameters used for the calculation of the charge state correction are those derived by Hubert et

al.' from their comparison with experiments. Measurements of the average energy loss of carbon

ions from 290 MeV/u to about 90 MeV/u are currently being made at NIRS . A preliminary com-

parison with the calculations shows agreement within 1 %. Several uncertainties are associated with

the algorithm used here. While estimates are given for these uncertainties , they may be modified as

more information becomes available.

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2.3.1. Mean excitation energies I

Most I — values are derived from experimental stopping power data for protons. Their uncer-

tainty therefore is related to that of the data. A reasonable choice might be an uncertainty in bet-

ween those given by ICRU report 49 and by Bichsel and Hiraoka" ) and Hiraoka et al"'13).

2.3.2. Shell corrections

The shell corrections are less than 1 % for most of the energies tabulated here. Their uncer-

tainty will contribute little to the total uncertainty.

2.3.3. Bloch and Barkas corrections

These corrections are also small for most of the energies listed here.

2.3.4. Charge state corrections

This correction is the dominant effect for the stopping power at small speeds. It may con-

tribute a total uncertainty of 30 % at 0.5 MeV/u. We used the correction method given by Hubert

et al.1).

2.3.5. Stopping power S

The overall uncertainty of S should be assumed to be about 2 % for T/u > 10 MeV/u and

may be as much as 30 % at very small speeds14). Nuclear collisions contribute much less than 0.1 %

to the energy loss, except at very small speeds.

2.3.6. Range R

For larger energies such as those used in cancer therapy, the uncertainty in R will be that of

S at high speeds because the influence of low energies on total range is quite small. For example,

the residual range of ions reduced to 10 MeV/u is only 0.5 mm. However, the overall uncertainty of

R also should be assumed to be about ± 2 %.

3. Tables

The independent variable for the tables is the ion energy divided by the mass number (i.e. 12

for carbon ions), and the unit is written as MeV/u. The stopping power is given in terms of the full

energy loss per unit thickness (here, MeV cm2/g).

The tables are given for a number of energies sufficient to give an error of less than 1 % for

other energies if an interpolation with the approximations

S = C Taand R = D TP

is made. This can easily be done on a pocket calculator which provides a linear regression routine.

One writes

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Table 2. Stopping power and range tables for carbon ion beam

Table 2. Continued

Table 2. Continued

and enters in T as the independent variable, in S or In R as its dependent function. The ,result of

the calculation then is S = exp (in S).

4. Conclusions

The calculations for the gases were made with the charge state correction used for the solids.

This assumption introduces additional uncertainties to the tabulated functions, which at the present

cannot be assessed. A problem which also must be considered in applications is the fragmentation

of the ions. It is considered in some publications, e.g. Shihver et al.15). For carbon ions at 290

MeV/u in water, near the Bragg peak, the dose due to fragments is about 6 % of the totall6).

Acknowledgement

The authors would like to thank Dr. T. Kanai for helpful suggestions.

References

1) Hubert F, Bimbot R and Gauvin H.: Range and stopping - power tables for 2.5-500 MeV/

nucleon heavy ions in solids, Atomic Data and Nuclear Data Tables, 46: 1-213, 1990

2) Walles R, Litton GM and Steward PG.: Stopping power, range, and terminal ionization of any

nucleus with 0.01 to 500 MeV/amu in any nongaseous material, including nuclear effects,

UCRL -19391, 1969

3) Sigmund P.: Light - ion stopping near the maximum, Nucl. Instr. Metho., B85, 541- 550, 1994

4) Brice DK.: Three - parameter formula for the electronic stopping cross section at nonrelativistic

velocities, Phys. Rev., A6, 1791-1805, 1972

5) Bethe H.: Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie, Ann.

Phys., 5, 325, 1930

6) Livingston SM and Bethe HA.: Nuclear physics, Revs. Mod. Phys., 9, 245-390, 1937

7) Bloch F.: Zur Bremsung rasch bewegter Teilchen beim Durchgang durch Materie, Ann. Phys.,

16, 285, 1933

8) Barkas WH, Birnbaum W and Smith FM.: Mass ratio method applied to the measurement of

L - meson masses and the energy balance in pion decay, Phys. Rev., 101, 778, 1956

9) Ahlen SP.: Theoretical and experimental aspects of the energy loss of relativistic heavy ionizing

particles, Revs. Mod. Phys., 52, 121-173, 1980

10) Bichsel H.: Stopping power of fast charged particles in heavy elements, NIST report ,

NISTIR -4550, 1991

11) Bichsel H and Hiraoka T.: Energy loss of 70 MeV protons in elements , Nucl. Instr. Metho.,

B66, 345 -351, 1992

12) Hiraoka T, Bichsel H et al.: Energy loss of 70 MeV protons in organic polymers , Med. Phys.,

99

20, 135 — 141, 1993

13) Hiraoka T, Bichsel H et al.: Energy loss of 70 MeV protons in tissue substitute materials, Phys.

Med. Biol., 39, 983-991, 1994

14) Bichsel H and Lubach A.: Neutron dosimetry with spherical ionization chambers. Part II: Basic

physical data, Phys. Med. Biol., 27, 1003, 1982

15) Shihver L, Tsao CH et al.: Total reaction and partial cross section calculations in pro-

ton — nucleus (Zt < 26) and nucleus — nucleus reactions (Zp and Zt < 26), Phys. Rev., C47, 1225 —

1236, 1993

16) Kanai T.: Private communication

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