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Calculations of charged-particle recoils, slowing-down spectra, LET and event-size distributions for fast neutrons and comparisons with measurements

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1979 Phys. Med. Biol. 24 18 (http://iopscience.iop.org/0031-9155/24/1/002) View the table of contents for this issue, or go to the journal homepage for more

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PHYS. MED. BIOL., 1979, Vol. 24, No. 1, 18-36.

Printed in Great Britain

Calculations of Charged-particle Recoils, Slowing-down SDectra. LET and Event-size Distributions for Fast Nhtrons and Comparisons with Measurements T. B. BORAK, P H . D . ~and T. G. STINCHCOMB, PH.D.$

t

Division of Biological and Medical Research, Argonne National Laboratory, Argonne, I L 60439, U.S.A. $ Department of Radiology, The University of Chicago,Chicago, I L 60637,U.S.A. Received 17 October 1977, infLnal form 4 X a y 1978 ABSTRACT.Arapidsystem has been developed forcomputingcharged-particle distributions generated in tissue by any neutron spectra less than 4 MeV. Oxygen and carbon recoils are derived from R-matrix theory, and hydrogen recoils are obtained from cross-section evaluation. Application to two quite different fission-neutron spectra demonstrates the flexibility of this method for providing spectral details of the different types of charged-particle recoils. Comparisons are made between calculations and measurements of event-sizedistributions for a sphere of tissue 1 pm in diameter irradiated by these two neutron spectra. LET distributions have been calculated from computedcharged-particle recoils and also derived from measurements using the conventionalapproximation thatall chargedparticles traverse the chamber. The limitations of the approximation for these neutron spectra are discussed.

1. Introduction

Any comparison of biological damage produced by neutron irradiation is complicated by the fact that neutron sources are rarely monoenergetic.Furthermore, the absorbed dose is delivered by charged particles indirectly produced from an uncharged radiation field. When thepath length of the charged particles is less than a few hundred +m, a detailed description of the chargedparticle spectrum becomes essential for dose determinations, especially near the interface of heterogeneous tissue components.It has also been demonstrated that the relative biological effectiveness (RBE) isrelated to linearenergy transfer (LET) (Elkind and Whitmore 1967, Hall 1973) and to the effective charge-to-velocity ratio (Zip)* (Katz 1970) of the recoil particles. From agiven neutronspectrum, we wish to computetheproduction of charged recoils withinavolume of tissue-equivalentmaterial composed of hydrogen,carbon,nitrogen and oxygen.Thespectra treatedhere consist predominantly of neutrons with energies below 4 MeV where collisions can be adequately described by elastic scattering and captureprocesses. Proton recoils are computed using a short table of differential elastic-scattering parameters

t Present address: Department of Radiology and Radiation Biology, Colorado State University, Fort Collins, CO 80523, U.S.A. $ Permanent address: Physics Department, De Paul University, Chicago, IL 60614, U.S.A. 0031-9155/79/010018+ 19$01.00

@ 1979 The Institute of Physics

Charged-particle Recoils and

LET

19

for hydrogen from the ENDF/B-IV neutron cross-section library. Since the other major constituents of tissue, namely carbon and oxygen, are both spinzero nuclei, a reduced R-matrix theory can provide a convenient and complete representation of elastic scattering. Since nitrogen nuclei do not have spin zero and are only a small componentof tissue, elastic scathering from these nuclei is simulated by replacing the nitrogen with carbon and oxygen, maintaining the original ratio of the latter two. At very low neutron energies, monoenergetic recoils are derived from H(n,y)d and N(n,p)C reactions using a simple l/w approximation. These approximations are suitable for radiation studiesinvolving the therapeuticuse of 252Cfimplants and neutron capture therapy. The methods are also useful for estimating biological damageresulting from accidental exposure to fission neutrons. Thecomputedcharged-particleproductionspectra for all thetypes o f recoils are converted into slowing-down spectra, using the stopping powers as functions of energy. These spectra are then mapped into functions of stopping power. If one neglects the removal of energy from the region of interest by delta rays, stopping power is equal to LET. Measurements to verify computations of dose distributions as a function of LET can be made using techniques based on proportional counters, solid state detectorsand nuclearemulsions.The use of aproportionalcounterhas advantages in this and similar applications since small spherical counters are readily available with a tissue-equivalent wall and may be filled with tissueequivalent gas of varying pressures to simulate tissue-equivalent spheres with diameters from one to several pm. Electronic pulse amplification is used with a multichannel analyser to give a frequency distribution of pulses, the sizes of which are proportional to the energy deposited in the chamber. This distribution is conventionally convertedto a distribution in LET by a transformation which assumes that all ionising particles start in the wall and completely traverse the gas cavity. Comparisons between measurements and computations will be presented for a modified fission neutron spectrum in air and a moderated fission spectrum located at the centre of an irradiated cube of water. These comparisons show important differences between the LET distributions computed directly and the LET distributions derived by the conventional transformation from the measurements. An alternative way of looking a t these differencesis to compare the measured distributions in event size, Y (i.e. energy deposition), with the distributions in event size transformed from the LET distributions computed directly. For the purpose of these comparisons, dose distributions in event size and LET are used rather than frequency distributions since the dose distributions are of primary interest. The difference between these computed and measured results is due (at least in part) to the inadequacy of the assumption upon which the conventional transformationisbased.TOdemonstratethis,event-sizedistributionsare derived directly from the computed recoil spectra, taking account of charged particles starting and stopping in the gas. These computations are then cornpared to the measurements. To the extent that theyagree, one concludes that

B.T.

20

Borak and

T . G. Stinchcomb

the simple and rapid methodof computing charged-particlerecoils is adequate, and that the distributions in LET computed directly aare closer approximation the to the true LET distributions than the LET distributionsderivedby conventional transformation from the measurements. 2.

Neutronspectra

I nt h e first comparison, charged-particle-recoil calculations andLET measurementsaremadeinairinside the high-flux-irradiation room of the JANUS reactor (Williamson and Frigerio 1972). The energy spectrum of this modified fission neutronsourcehasbeenpreviouslydeterminedinair at representative locations using a proton-recoil spectrometer (Bennett and Yule 1972), activation detectors and a helium-recoil spectrometer (Frigerio 1971). Very good agreementamong the varioustechniques was obtained.The spectrum shows a peak a t 0.8 MeV with 96% of the neutrons below 4 MeV. Measurements with gold foils show that the thermal neutrons (underCd) have a kerma rate less than 0.02y0 of the fast-neutron kerma rate. The dose rate due to gamma rays in the empty room was determined to be less than 3% of the totaldose using paired ionisation chamber techniques (Williamson, Frigerio, Holmblad, Trier and Johnson 197 1). Comparisons between calculations and measurements were also performed with a moderated fission-neutron spectrum at the centre of a cube of water (of 16 cm side) placed in the high-fluxroom.This neutronspectrum was generated from the spectrum in air using a Monte Carlo neutron transport code (Frigerio 1973). The version of the code used did not trace charged particles. Pseudo spectrometers recordthe passage of neutrons a t desired locations within the phantom. Convergence of this spectrum is considerably more rapid than the collection of eventsnecessary for dose determinations. Using afirst collision approximation with published kerma conversion factors (Bach and Caswell 1968, Ritts and Solomito 1968),the computed kerma rate as a function

Fig. 1. Neutron energy spectra in JANUS reactor facility. Solid line is spectrum measured in air. Dashed line is spectrum computed in centre of 16 cm cube of water. Area under each curve is normalised to one neutron per cmz.

Charged-particle Recoils and LET

21

of depth along a central axis of the phantom agreed very well with ionisation 1977, personalcommunication).Fig. 1 chambermeasurements(Holmblad shows the JANUS reactor spectrum measured in air and that computed at the centre of the 16 cm water cube. 3. Charged-particleproductionspectra Once a neutronspectrumisdetermined,computation of charged recoils becomes an exercise in integrating differential cross-sections. I n this representation, elasticscatteringisassumed to be atwo-body process with the target nucleus a t rest in the laboratorysystem. Thespectraldistribution of recoil particles(integrated over the neutron energies) is obtained using

Np(Ep) dE, = rrN dE,

jdEn '(En)

+

(M, M,)2 do M, M,E, dsz (En, 6')

where

NP(EP) dE,

E, E, M, M, 6'

N do/dR dR

= the number of charged particles (of a particular ion type,

produced per gram of medium with energies between E , and E , +U,, = energy of the target nucleus recoiling in the laboratory system, = energy of the incident neutron in the laboratory system, = mass of the neutron, = mass of the target nucleus, = neutron scattering angle in the centre of mass system, = differential neutron fluence (neutrons MeV-1 cm-2), = nuclei per gram for the target= N,/A, = differential cross-section in the centre of mass system, and = solid angle in centre of mass system.

The cross-section for neutron-hydrogenelasticscatteringisasmooth function of energy. For this reaction, we use a table of cross-section against energy consisting of 92 entries as provided by ENDF/B-IV. The data points aredistributed to facilitatea simple log-log interpolation between entries. Although the scattering is nearly isotropic in the centre of mass, there is a slight asymmetry above 0.5 MeV. This asymmetry is incorporated in tables of angular distribution coefficients provided by ENDF/B-IV. The heavier elements of tissue-like material have more complicated crosssection structures which may be dominated by resonantpeaks.Adetailed tabularrepresentation of thesestructures becomes quite cumbersome. I n certain situations theoretical formulations can dramatically reduce both the computation and input data (from several reels of magnetic tape to a deck of FORTRAN statements less than three inches thick) with no loss in accuracy. Any neutron spectrum with energies below 4 MeV is suitable for such a formalism, The major step in any nuclear theory is the derivation of a collision matrix. From the collision matrix one can compute observable quantities such as total

T.B . Borak and

22

T . G . Stinchcomb

cross-section (U) and differential cross-section (daidfi). The principles involved for deriving a collision matrix from R-matrix theory have been extensively reviewed byLane and Thomas (1958). Below neutron energies of 4 MeV scattering from 12C and 13 ' 0 is dominated by elastic processes. Furthermore since both are spin-zero nuclei, only one spin channel isopen and the R-matrix can be reduced to a single R-function. One basic assumption of the theory is the existence of a critical interaction radius, beyond which the neutron-nucleus interaction vanishes. The fundamental R-function is then obtained from the following boundary conditions : aYyl(dYz/dr)r=a = (1+ BR)/R where

B = the boundary condition constant, R = the R function, U = channel radius, l = total orbital angular momentum, and Yz= the radial part of the Schrodinger wave function for r

a such that

Here, k is the neutron wave number given in fm-l by

k

277

= -=

h

0.21968

( FA+) 1

En1f2

where A is the atomicmass of the targetnucleus and E, is the incident neutron energy in MeV (in the laboratory system). The R-function has explicit energy dependence given by

E,,, are the energies of the resonance levels denoted by X, and the yZnzJ are the reduced widths of these levels for orbital angular momentum l and total angular momentum J . Using this R-function and the solutions to the wave equation, the collision matrix qJ can be expressed as a function of RzJ. For the case of neutronscattering from a spin-zero target, the collision function is diagonal in spin, and the requirement that the collision matrix be unitary yields qJ = exp (2iSlJ) (6) where S,, are the scattering phase shifts. I n this situation the total angular momentum has only two possible values : J = l + 8 ; J = 1 - 8. Once the phase shifts have been determined, it is possible t o compute the non spin-flip and the spin-flip amplitudes.Thedifferential cross-section inthe centre of mass is expressed as the sum of the squares of these scattering amplitudes. The total cross-section is the integralof the differential cross-section over all solid angles. R. Holt hasrecently reported an R-matrixanalysis of n-12C elastic scattering below 4 MeV (Holt, Smith andWhalen 1975) and n-la0 elastic scatteringbelow

Charged-particle Recoils and LET

23

4 MeV (Hickey, Pirk, Holt and Nath 1974). We have used a modified version of a computer code developed by Holt tocompute the differential cross-section forelasticscattering from carbon and oxygen.TheR-functionparameters used inthis analysis are givenin the Appendix.Thismethodprovidesa complete representation of the elastic scattering process and rapidly yields a detailed spectrum of charged particles produced a t all incident energies and scattering angles. For elastic scattering, nitrogencollisions are ignored and the amounts of carbon and oxygen are adjusted to compensate for the missingnitrogen. Inthethermalandepithermal energyrange, neutronscattering from hydrogen and nitrogen is dominated by capture processes. Neutron capture by hydrogen H(n, y)d has a Q-value of 2.226 MeV yielding a monoenergetic deuteron of 1.32 keV. The cross-section has a value of 0.332 barn a t a neutron energy of 0.0253 eV (2200 ms-l) and is a smooth function of energy with a v-l dependence. Nitrogen capture I4N(n, p)14C has a Q-value of 636 keV with the excess energy shared by a 584 keV proton and a 42 keV carbon recoil. The cross-section is 1.82 barn a t 0.0253 eV with a v-l dependence. Proton, deuteron and carbon recoils from these captures are added to the appropriate production distribution obtained from elastic scattering. For both elastic and capture processes we haveassumed that the detector does not disturb the neutron spectrum, and in effect these are first collision processes. Fig. 2 shows the energy distribution of recoils for the JANUS fission neutron spectrum in air. The energy distribution of elastic recoils for the moderated neutron spectrum in water is shown in fig. 3. For this spectrum, the capture processes yield 0.063 recoil protons and carbon nuclei per gram of nitrogen and 0.159 recoil deuterons per gram of hydrogen.

4. Slowing-down spectraand LET distributions

The slowing-down(or equilibrium) spectrumof chargedparticles is the spectrum obtained in a medium in which the charged particles are produced homogeneously and at a constant rate if the dimensions of the medium are larger lot

:

10

i

Carbon r e c o l l s

,

"

'

"

"

,

I

Oxygen recoils

;

'i\

1 0 " " " ~ ~ . 0 0.4 0.8

\

t 12

10-1*+

0

Energy (MeV1

Fig. 2. Charged-particleproductionspectra from the JANUS modified fission-neutron spectrum in air. Each curve gives the number of recoils per gram of element per neutron cm-2.

T.B. Borak

24

-

! L z L J

'O

lOL/=LJ

T.G . Xtinchcomb

and

j '1, : ! a \ 4

2 I

102

i

10 -1;

I

1 :\",

\

._

1

10-2;

j

1U3j

1

' ! 7 1

IO"!.

>

' C 10'2:

2%10": 1

ct

l

i 1 o-L

0

.

16

32

L.8

10".

,

0

.

OL

,

-~

08

Energy IMeV

12

-~~.

10-5"-

0

028

056

08L

l

Fig. 3. Charged-particle production spectra from the moderated neutron spectrumat the centre of a 16 cm cube of water. Each curve gives the number of recoils per gram of element per neutron cm-2.

than the longest range of any of the charged particles produced (Roesch and Attix 1968). Here the wall thickness of the detector is 1270 pm compared to a range of about 350 pm for a 5 MeV proton and is assumed sufficient to establishequilibrium.The slowing-down spectrum is usuallyobtainedfrom the production spectrum by use of the approximation that the particles lose energycontinuously.Thisis the continuous-slowing-down approximation (Roesch 1968) and is assumed to be validin this application. The relation between the production spectrum and the slowing-down spectrum under these conditions is given by Caswell and Coyne (1972)

where

#(E,)= thestopping

power, dE/dx,in MeV per gcm-2 (for this ion type) atenergy E, (also in MeV), N,(E,) dE, = the numberof charged particles (of this ion type) which cross a unitarea (cm2) in the medium having slowed down t o energies between E, and E,+dE, (in MeV), and E, = the maximum energy a t which charged particles (of this ion type) are produced. The tables of stopping powers and ranges which are used here are for tissue (0.102 hydrogen, 0.123 carbon, 0.035 nitrogen, 0.740 oxygen, by weight). They were obtained through the courtesy of R. S. Caswell of the U.S. National Bureau of Standards. The derivation of these tables from experiments and theories has been outlined (Caswell and Coyne 1973). Although the composition of the tissue-equivalent materials used for the wall and gas of the proportional counter isdifferent in the amountsof carbon and oxygen, thesedifferences have only a small effect on the stopping power and ranges and are not taken into account in the computation of slowing-down spectra from production spectra.

LET

Charged-particle andRecoils

25

These stopping powers in tissue areshown in fig. 4 for proton, deuteron, carbon and oxygen recoils. Theresulting slowing-down spectra for productionin tissue-equivalent plastic (Smathers, Otte, Smith, Almond, Attix, Spokas, Quam and Goodman 1977) (0.102 H, 0.834 C, 0.064 0 , by weight: the nitrogen(0.036) and the other elements (0.035) were divided between the carbon and oxygen retaining the ratio of the latter two) are shown in figs 5 and 6. - ..

.

" a

x103

.Q

l2

.. .

.

12CO Ceuteron

1200 Proton

x1

o3

l2

Carbon

-

~.

Nltrogen

-

0

,'A,

$ i 8 L ~

" '

8 L

/'

. "

10-2

102

1

O1o-L

18' Oxygen

II

12'

..

~

,

.

i

-~ I ' .

0

10'

10-2 1 Energy l MeV i

. ..

-.

',

6: .

olo-t

i

,

-

-

.-

10"

..

1

10-2

.

.

102

Fig. 4. Stopping power in tissue for recoil charged particles.

"E 10-3r U

& Q

21

0 :

!

.

m qo-5: -

105{

1 *;

.cl 8..

19"

_ I

,,,,> t

Frctons

4

G i/" z : L. 13 ._

lo"7

.__ -_ L _ -II.--

:

i / ' '

i

""p1lo-

. 102

1

.

.-

. .._L

Carbon ,,.- \t "

\

j 10"

-7 1

1

i Oxygen

10-7;

~

10 1 E w r g y (MeV1

/'II

1

1

,-,'

~10-gy"'

~

i !-

10"

II

1

1

1

"

"

~

1o-2

I l

Fig. 5 . Slowing-down spectra for charged recoils producedin tissue-equivalent plastic fromthe modified fission-neutron spectrum in air. Normalised to one neutron per cm2.

The slowing-down spectrum is the basis for the frequency distribution in LET. Only a change in independent variable from energy to stopping power needs to be made. Hence Ns(E,)dE, = NL(S)dX (8) where NL(S)dS is the number of particles crossing a unit area (cm2) in the medium with LET values between S and S + dS. The small difference between LET and stopping power, S , due to energy removal from the region of interest by delta rays, is being ignored. Thus LET is equated to S and for ease in computation is given the same units (MeV per gcm-2). Upon presentation of results, the unit for LET is changed to keV pm-1 (1 keV pm-1 = 10 MeV per g cm-2 for EX density of unity) and the notationis changed from S to L.

T . B. Borak and T . G. Stinchcomb

26 10-)I

Protons

;

i

10-6: .

!

-~ -

" " i

Deuterons

i

Fig. 6. Slowing-down spectra for charged recoils produced in tissue-equivalent plastic at the centre of a 16 cm cube of water from the moderated neutron spectrum normalised to a fluence of one neutron per cm2. The dashed curves represent t h e contributions from neutron-capture events.

Although S is a single-valued function of E , the reverse is not the case; and for some values of X there are contributions from two (or more for C, N and 0 recoils) values of E . This can result in sharp cut-offsof the frequency distributions in LET (e.g. for protons as the LETis increased through 1000 MeV per g cm-2). Sharp increases can also result (e.g.for protons, asthe LETis increased through 150 MeV per g cm-2 suddenly adding the contributions from large numbers of low energy protons). I n addition, large peaks can occur (e.g. for carbon recoils, as the LET is increased through 1700 MeV per gcm-2 where between the S against E curve is nearly parallel to theE axis). Thus the relation the intervalsizes d E a n d dinS eqn(8) is important. The slowing-down spectrum which is tabulated for the same values of energy used in the stopping power table (6.1y0spacinginenergy) is mappedinto the desiredstoppingpower interval (4.7% spacing). The spectrum is linearly interpolated when the latter interval does not completely overlap the interval between successive values in the stopping power table. Absorbed dose distribution in LET can be simply related to the frequency distribution by DL(S)dX = SNL(S)d S (9) where DL(S)d S is the dose in MeV 8-1 due to particles with LET between S and S + dS. Thus the unit of DL(#) is MeVg-l per MeV (g-l cm2), or simply cm-2. of results Thisunit is changed t o rad per keVpm-luponpresentation (1 cm-2 = 1.602 x 10-7 rad per keV pm-l for unit density). Fig. 7 shows calculated dose distributionsin LET for the twodifferent neutron spectra of interest. The ordinates in these semi-logarithmic graphs have been multiplied by L in order to preserve the ratios of areas under the

LET

Charged-particle andRecoils

27

curves (i.e. LD,(D) is plotted against the logarithm of L ) . I n addition the curves have been normalised so that the total dose in each case is unity (i.e. J'D,(L) dL = JLD,(L) d(ln L ) = 1 rad). Especially notable isthe clear separation between the distributions a t higher LET due to heavy-ion recoils and those a t lower LET due to proton and deuteron recoils. ,

"