J ,

/

NASA Technical Memorandum 4753

Multilayer Analyses of an Iron Radiation

Beam Experiment

i

J. L. Shinn and J. W. Wilson

Langley Research Center • Hampton, Virginia

E. V. Benton

University of San Francisco • San Francisco, California

F. F. Badavi

Christopher Newport University • Newport News, Virginia

National Aeronautics and Space Administration

Langley Research Center • Hampton, Virginia 23681-0001

April 1997
Acknowledgments

E. V. Benton would like to thank Walter Schimmerling of NASA

Headquarters for suggesting our involvement in the fragmentation

work and Jack Miller of UC-LBL and his group for sharing their

Bevalac beam time and for their help during the irradiations. The portion of the work performed at the University of San Francisco was

supported partially by NASA grants NAG9-235 and NAGW-4154.

Available electronically at the following URL address: http://techreports.larc.nasa.gov/ltrs/ltrs.html

Printed copies available from the following:

NASA Center for AeroSpace Information

800 Elkridge Landing Road

Linthicum Heights, MD 21090-2934

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-2171

(703) 487-4650
Abstract

A nonperturbative analytic solution of Green's function for heavy

ion transport in a single medium is extended to multilayer transport

media. This extension is implemented in the Green's function code

(GRNTRN), which is being validated with laboratory experiments.

Good agreement is achieved between an iron radiation beam experiment and GRNTRN only when the interactions of the iron radiation

beam with several important layers of material located upstream of the

target within the beamline are considered.

Introduction

Astronauts and occupants of high-altitude aircraft are exposed to heavy-ion cosmic radiation that

originates from the Sun and galactic sources. The shielding and exposure of these individuals are

controlled by the transport properties of the radiation through the specific materials. Efficient spaceradiation-transport computer codes (refs. 1, 2, and 3) have been developed and applied to a wide range

of missions, but the results of these codes could not be validated accurately in laboratory experiments

(ref. 4). Recently, the use of a nonperturbative approach to Green' s function solutions has led to an efficient laboratory code, namely, Green's function transport (GRNTRN) code (ref. 5), which will be further developed for space-radiation-transport calculations and validated with laboratory experiments.

Preliminary validation of the laboratory Green' s function code (GRNTRN) has been made by comparing the theoretical results with recent iron radiation beam measurements made at Lawrence Berkeley

Laboratory Bevalac accelerator by Shinn et al. (ref. 6). The iron beam had been accelerated to

600 MeV/amu and passed through a series of beam-transport elements, triggering devices, and a leadfoil beam spreader prior to emerging from the beam tube and striking a layer of aluminum or polyethylene targets. Overall, fairly good agreement between measurements and predictions from GRNTRN was

reached when the nuclear database used in GRNTRN was improved in the physical formulation and

corrected for mass and charge conservation (refs. 7 and 8). Furthermore, as the code was modified to

include the interactions of beam and lead foil, better agreement was reached except that the predicted

fragmentation was still low by 30 percent for the aluminum target. The code modification was made to

extend the nonperturbative solutions for a single medium to inhomogeneous, two-layer transport media

(for lead foil and target material). In this paper, we present further improvement to our analytical results

by modifying GRNTRN to include interactions from other significant layers of material from the beam

tube. The development of the nonperturbative method to inhomogeneous, multilayer (three or more layers) transport media will be given with a brief review of the analytic properties of the perturbative solution of the heavy-ion-propagation problem and a nonperturbative approximation to the full perturbation

series.

Transport Equation

We restrict our attention to the multiple charged ions for which the Boltzmann equation may be

reduced (ref. 9) to

k

where d_j(x,E) is the flux of ion typej at x with energy E (in MeV/amu), Sj(E) is the change in E per unit

distance, (_j is the total macroscopic reaction cross section, and ojk is the macroscopic cross section for
w

collision of ion type k to produce an ion of type j. The solution to equation (1) is to be found subject to

the boundary condition

_)j (0,E) = fj (E) (2)

The corresponding Green's function is introduced as a solution of

_ESj(E) + CY Gjm(X, E, E') = Z_jkGkm(X, E, E') (3)

k

subject to the boundary condition that G jm(O, E, E') = _jm_)(E - E').

The preceding equations can be simplified by transforming the energy into the residual range as

and defining new field functions as

rj = fE dE'

aO Sj(E')

_j(x, rj) = Sj(E)d_j(x,E)

Gjm(X , rj, rm) = Sj(E)Gjm(X, E, E')

}j(rj) = _Sj(E)Oj(O,E)

Gjm(X , r j, rm) = _jkGkm(X, rk, rm)

-- - Or----j+ (YJ

Equation (3) becomes

with the boundary condition

= Z_/Aj.

# •

Gjm(O , rj, rm) = 8jmS(r j- rm)

and with the solution to the ion fields given by

_Ij(X, _oo # ^ # # r j) = Z 0 Gjm(X' rj, rm)fm(rm) dr m

m

Note that vj is the range scale factor such'that vjrj = Vmr m and can be taken as vj

Perturbative Green's Function

The solution to equation (8) is written as a perturbation series:

Gjm(X, r j, rm) = _"z__ G(i)jm, '(xr j, rm)

i

where

G(0). r' - '

jmtX, rj, m) = exp(-cyjx)_jmS(X + rj rm)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)
and the first collision term is

Vj(Yjm

G(1)tx rj, rm) - ...,exp(-(Ijxj (_mXm)

jm" ' IVm- vj I

t

where G(1)fx jm " ' rj, rm) takes on nonzero values for

vj v. --(r;+x) " <--2-Jr.+x

Vm J < r m_ Vm J

and

(13)

(14)

xj = (VmX + vjrj - Vmrm)/(v m - v j) (15)

X m = (Vmr m - VjX - Vjrj)[(V m - Vj) (16)

Note that in equations (13)-(16), x.j -I-Xm = x for all rj and r'. The significance of X m is that it is the

distance ion m traveled from the boundary to the collision site at which the ionj was produced and must

now travel distance xj before reaching x. Similar results hold for higher order terms (refs. 10 and 11).

p

Whenever rj,r m < x, it is relatively easy to show that

rju tl_

rjl G)m)(X, rj, rm)dr j = _jrng(J' m)

(17)

where the integral limits are the interval limits in equation (14)and g(j, m) is the g-function of two arguments defined by the following relations:

g(j) = exp(-erjx) (18)

and

g(Jl' J2 ..... Jn' Jn+l) =

g(J l' J2 ..... Jn-l'Jn) -g(Jl' J2 ..... Jn-l'Jn+l )

(_jn+l -- (Yjn

(19)

The first collision Green's function term may be approximated by the spectral averaged value as

G(1), rj, r" 1 Ir ju (1) _ rj, rm) drj

jm (x, m) = -- jl Gjm (_' rju -- rjl

_ Vj(Yjmg(j, m)

m

XlVm-Vjl

(20)

and is functionally dependent only on depth of penetration x. Its contribution applies only over the

energy range of equation (14) and is null outside. The higher order terms are approximated by their

spectral averages as

Vj(YJkl(Yklk2...(Yki_lmg(j, k 1, k 2..... ki_ 1, m)

(i) _ rj, Gjm(_' rm)= Z xlvm_Vj [

k1,k2..... ki_1

(21)
i P • • •

Note that G_i)m (x, r j, rm) IS purely dependent on x for z > 0, and we represent the expression as Gjm(X) (i)

(ref. 4). In terms of equations (11)-(21), the solution to equation (1) becomes (ref. 4)

r

^ [_I I ^ I

_lj(x, r j) = exp(-_jx) f j(r j + x) + Z G}/m)(X) L m(rml ) - Fm(rmu)?

m, i

p p p

In equation (22), rm, u and rm, l are given by the upper and lower limits of the rm,

^

equation (14). The symbol Fm(rm) refers to the integral spectrum

(22)

satisfying

Fm(rm) A = fm(r) dr (23)

Nonperturbative Green's Function

We now introduce nonperturbation terms for the summation in equation (22). First, we note that

g jm (x) = _jmg(m) + _jmg(J, m) + Z Ojk_kmg(j, k, m) +...

k

(24)

A term-by-term comparison of the series (22) with the series (9) shows that, in the spectral average

approximation, we may write the approximate solution as

^ vVj[gJm(X)--exp(-_JX)_jm]IFm(rmu)-lTm(rml) 1

Vj(x, rj) : exp(-(Ijx)f(rj+x)+ _m xlvm-V:l (25)

which is a relatively simple quantity once gjm(X) is evaluated (ref. 4). The advantage of equation (23)

is that gjm(X) satisfies the convolution product relation (ref. 12) as

gjm (x) = Zgjk(X- Y)gkm(Y) (26)

k

for any positive values of x and y. A solution for small x is easily obtained from the first few terms of

equation (24). Equation (26) may then be used to propagate the function gjm(X) over the entire solution

space, taking arbitrarily large steps.

Nonperturbative Method for a Shielded Medium

The major simplification in the Green's function method results from the fact that the scaled spectral distribution of secondary ions to a first approximation depends only on the depth of penetration, as

seen in equations (20), (21), and (22). Our first approach to a multilayered Green's function will rely on

this observation and assume its validity for multilayered shields.

Consider a domain labeled as 1 that is shielded by a second domain labeled as 2; the number of type

j ions at depth x in 1 due to type m ions incident on domain 2 of thickness y is

gl2jm (x' Y) = Z gljk (x)g2km(y) (27)

k

The leading term in equation (24) is the penetrating primaries as

g12jm( X, y) = exp(-61jx- _2jY)Sjm + [g l2jm(X, y) - exp(--CYljX- q;2jY)_)jm ] (28)

,,;
where all higher order terms are in the bracket of equation (28). The first term of the scaled Green's

function is then

G (0) 12jmt rx ' Y' rj, rm) = exp(-cljX - (Y2jY)_jm_[X + rj- (r m - py)] (29)

where p is the range scale factor for the two media

p = Rlj(E)/R2j(E ) (30)

which is fairly constant above 10 MeV/amu or so. We take a single value for p that corresponds to

600 MeV/amu. The secondary contribution is similarly found from equation (14) as

vj vj

_mm(rJ + x + PY) -<r'm_VmJ<--r. + x + py (31)

from which the average spectrum is evaluated. The full approximate Green's function is then

G12jm(X, y, rj,rm) = exp(-oljX - (Y2jY)_jm_(X + py + rj - rm)

+ Vj[gl2j m(x' Y) - exp (--(YljX- (Y2jY)_jm]

(x + Oy)lVrn- Vjl

(32)

Equation (32) is our first approximation of Green' s function in a shielded medium of two layers, and is

easily modified to multiple layers. (See appendix.)

For the first spectral modification, the first collision term has the properties

[ VjO l jm eXp (- a lmX - 152mY )

cr12jm£X, y, rj, rm) = I VJlYzjm eXp(--_l---_ x- (Y2jY)

L IVm-vjl

r t _ r p }

m mu

t p

r m rml

(33)

These properties are used to correct the average spectrum to

V .(1) rx

G (1) (x, y, rj, r" j612jm_ , Y)

12jm m) = (X+ py)lVm-- Vj]+ bjm(X, Y)(r m - _'m)

(1) . ,

where gl2jm£X, y) is the first collision term of equation (32) and

(34)

• p

r mu + rml

_m - 2 (35)

is the midpoint of rm between its limits given by equation (31). The bjm term of equation (34) has the

property that

frmuh ( /

Jrml _jm,X, y)(r'- m) dr" = 0

(36)
whichensuresthatthefirsttermof equation(34)isindeedtheaveragespectrumasrequired.Thespectralslopeparameteris

bjm(X, y) = V jV m[°ljm eXp(- (Ylm x - (Y2m y) - (Y2jm eXp(-(Ylj x - (Y2jY)]

(x + py)(Vm - vj)lVm -- Vj[

(37)

A similarly simple spectral correction could be made to the higher order terms. The spectral correction

given in equation (37) is included in the present Green's function code.

Spectra for Laboratory Beams

The boundary condition appropriate for laboratory beams is given by

1 V-(E-Eo) 2-

fj(E)- :exp/ 7_

,,/27ro [ 20

(38)

where E 0 is the mean energy of beam with an energy spread 6. The cumulative spectrum is given by

F j(E) = _[1 _ erf|----m---)l fE-Eo'q ',,42o _1

The cumulative energy moment needed to evaluate the spectral correction is

Ej(E) = _E 0E1-erfl---_---H\ fE_/2c_ - E0-JJq + _t_ exp E

The average energy on any subinterval (El, E2) is then

(E - E0 )2-

The beam generated flux is

Iltj(X, y, rj)

2o 2

(39)

(40)

= F-'j(E1)-Ej(E2) (41)

Fj(E1) - Fj(E2)

exp(- (_ljX -- (_2jY)?j(rj + x + py)

+ _G_i)m(X,y)[Fm(rmu)-Fm(rml)]

m, i

+z_-" b(1)(Xjm, ' Y)[rm(E) - rm] [Fm(rmu) - 17m(rml )]

m

(42)

where E is evaluated with equation (41), with E 1 and E 2 as the lower and upper limits, respectively,

associated with rml and rmu.

The linear energy transfer (LET) distributions behind aluminum and polyethylene targets were

measured with a stack of four CR-39 plastic nuclear track detectors (PNTD's) placed behind the target

(ref. 13). The incident beam intensity was monitored by a thin layer of CR-39 PNTD placed immediately in front of the target. Calibration of these detectors with various ion beams of known LET resulted

in a response function that is approximately Gaussian with a LET-dependent width. The detectors and

targets were run in good geometry so that acceptance corrections are not required. The detector response

function is used to compare the experiment with the calculated LET distributions from GRNTRN,

assuming 0.2 percent beam-energy spread in equation (38).
Figuresl(a) and(b) showmeasuredandcalculatedLET spectrabehind2-g/cm2 aluminumand

4.6-g/cm2polyethylenetargets,respectively.Thecalculationsincludetheinteractionsbetweentheiron

beamand2.24g/cm2of the"thin" lead-scatteringfoil placedupstreamof thetarget.Theenergyof the

ironbeamprior to its interactionwiththetwo-layerlead-targetcombinationwas557MeV/amu.The

calculatedLET spectrafor fragmentsarein goodagreementfor thepolyethylenetargetexperiment

(fig. l(b)),butare30percentlowerthanthemeasurement forthealuminumtarget(fig. l(a)).Notethat

thespreadineachisotopepeakresultsmostlyfromthedetector-response functionratherthanthebeamenergyspread.

Someof thematerialsusedinthebeamline,otherthanleadfoil, thatcouldcausesignificantmodificationsto thebeampriorto its interactionwiththetargetwerefurtheridentified.In ourlatestcalculation,a0.16-cm-thick(0.43-g/cm2)aluminumexitplateandtwopolystyrenescintillatorsof 0.16-cmand

0.32-cmthickness(total0.49g/cm2)wereincludedin ourtransportanalyses.Equations(42)and(32)

weremodifiedfor a four-layer(leadfoil, aluminumexitplate,polystyrene,andtarget)configuration

andimplementedin GRNTRN.Theimprovedresultfor the 2-g/cm2 aluminumtargetappearsin

figure2.Forthepolyethylenetarget,theresultsof afour-layercalculation(notshown)werenotappreciablydifferentfromthoseof thetwo-layercalculation,aswasexpected.Becausethefragmentationfor

thepolyethylenetargetisfargreaterthanthoseforthealuminumtarget,thecontributionsfromthealuminumexitplateandpolystyrenearefarlesssignificantthanthosefromthelead-foilandpolyethylene

target.Thebeamenergyforthefour-layershield-targetcombinationwasinferred,takingintoaccount

theenergylossthroughtheselayers,to be583MeV/amu.Thelatterenergyvalueis muchcloserto the

beamenergyat extraction(600MeV/amu),whichindicatesthefour-layerconfigurationis a good

approximationto theexactone.

Concluding Remarks

The iron radiation beam experiment with CR-39 plastic nuclear track detectors (PNTD's) provides a

fairly accurate laboratory validation of Green' s function transport (GRNTRN) code. Through the multilayer analyses, it is concluded that for thinner targets, the effect of lead-scattering foil, beam-transport

elements, and triggering devices in modifying the characteristics of the beam is rather significant.

Laboratory validation up to this point provides confidence in the validity of the Green's function

method and, in an approximate sense, in the correctness of the nuclear physics theory. Any refinement

in the nuclear database used in GRNTRN ultimately will depend on better resolution of each isotope

measurement with a state-of-the-art detector system (Zeitlin et al., Radiat. Meas., vol. 23, no. 1, 1994)

and more sophisticated analyses, including multilayer configuration.

NASA Langley Research Center

Hampton, VA23681-0001

December 16, 1996

• 7 -'
Appendix

Derivation of Approximate Green's Function for Three-Layer Configuration

The formalism presented in the preceding text is extended to a three-layer configuration as follows.

The solution to equation (26) in a three-layered medium is

g123jm(X, y, Z) = Zgljk(X)g2kl(y)g3lm(Z) (A1)

kl

The leading term is the penetrating primaries, and equation (A1) may be written as

g123jm(X, y, z) = exp(-_ljX - ¢_2jY - _3jZ)_jm

+ [g123jm(X, y, Z) - exp (OljX - 1_2jY - IJ3jZ)_jm]

The scaled Green's function is then

(A2)

where P2 = Rlj(E)/R2j(E)

becomes

G123jm(X, y, z, rj, rm) = -exp(-oljX - o2jY - o3jz)_jm _(x + P2Y + P3 z + rj - rm)

-t v j[g123jm(X' y' z) - exp(-_ljX - _2jY - _3jz)_jm ] (A3)

(x + P2Y + P3z)(Vm - v j)

and P3 = Rlj(E)/R3j(E)" The range condition of equation (17)

vj vj

..--z-(r, + x + P2Y + P3 z) < rm < --r. + x + P2Y + P3 z

Vm J -- Vm J

(A4)

The spectral corrections are similarly derived.
References

1. Wilson, John W.; Townsend, Lawrence W., Schimmerling, Walter; Khandelwal, Govind S.; Khan, Ferdous, Nealy, John E.;

Cucinotta, Francis A.; Simonsen, Lisa C.; Shinn, Judy L.; and Norbury, John W.: Transport Methods and Interactions for

Space Radiations. NASA RP-1257, 1991.

2. Shinn, Judy L.; and Wilson, John W.: An Efficient HZETRN (A Galactic Cosmic Ray Transport Code). NASA TP-3147,

1992.

3. Wilson, John W.; Chun, Sang Y.; Badavi, Forooz E; Townsend, Lawrence W.; and Lamkin, Stanley L.: HZETRN: A Heavy

Ion/Nucleon Transport Code for Space Radiations. NASA TP-3146, 1991.

4. Wilson, J. W.; and Badavi, E F.: New Directions in Heavy Ion Shielding. Proceedings of the Topical Meeting on New

Horizons on Radiation Protection and Shielding, American Nuclear Soc., Inc., 1992, pp. 198-202.

5. Wilson, J. W.; Shavers, M. R.; Badavi, E E; and Miller, J.: Nonperturbative Methods in HZE Propagation. Radiat. Res.,

vol. 140, no. 2, 1994, pp. 241-248.

6. Shinn, J. L.; Wilson, J. W.; Badavi, F. F.; Benton, E. V.; Csige, I.; Frank, A. L.; and Benton, E. R.: HZE Beam Transport in

Multilayered Materials. Radiat. Meas., vol. 23, no. 1, 1994, pp. 57-64.

7. Shinn, Judy L.; Wilson, John W.; Benton, E. V.; Csige, I.; Frank, A. L.; and Benton, E. R.: LET Spectra of Fragmented

600 MeV/A Iron Beam on Aluminum and Polyethylene--A Preliminary Analysis. NASA TP-3436, 1994.

8. Wilson, J. W.; Shinn, J. L.; Townsend, L. W.; and Tripathi, R. K.: NUCFRG2: A Semiempirical Nuclear Fragmentation

Model. Nucl. Instrum. Methods in Phys. Res. B, vol. 94, 1994, pp. 95-102.

9. Wilson, John W.: Analysis of the Theory of High-Energy Ion Transport. NASA TN D-8381, 1977.

10. Wilson, John W.; Townsend, L. W.; Lamkin, S. L.; and Ganapol, B. D.: A Closed-Form Solution to HZE Propagation.

Radiat. Res., vol. 122, no. 3, June 1990, pp. 223-228.

11. Wilson, John W.; Townsend, L. W.; Bidasaria, H. B.; Schimmerling, Walter; Wong, Mervyn; and Howard, Jerry: 2°Ne

Depth-Dose Relations in Water. Health Phys., vol. 46, no. 5, May 1984, pp. 1101-1111.

12. Wilson, John W.; Costen, Robert C.; Shinn, Judy L.; and Badavi, Francis F.: Green's Function Methods in Heavy Ion

Shielding. NASA TP-3311, 1993.

13. Benton, E. V.; Ogura, K.; Frank, A. L.; Atallah, T.; and Rowe, V.: Response of Different Types of CR-39 to Energetic Ions.

Nucl. Tracks Radiat. Meas., vol. 12, 1986, pp. 79-82.

_:
10-1

-- GRNTRN

._ o Experiment _

i 0-3 g e_ So_ _

10-4 milii_HVr, ,,. ..... _......... _ _o

0 50 100 150 200 250

LET, keV/_tm

(a) 2.24 g/cm 2 Pb + 2 g/cm 2 A1.

100

__ 10-1 l

10-2

O

10-3

10-4

0

--GRNTRN

o Experiment

o o

50 100 150 200 250 300

LET, keV/_m

(b) 2.24 g/cm 2 Pb + 4.6 g/cm 2 polyethylene).

Figure 1. Transport measurements and calculations for lead and target (two-layer) combination for

557-MeV/amu iron beam.

100

10-1

_ 10_2

lO-3 r_

10-4

0

-- GRNTRN

o Experiment

o o

50 100 150 200 250

LET, keV/gm

Figure 2. Transport measurements and calculations for four-layer shield and target combination

(2.24 g/cm 2 Pb + 0.42 g/cm 2 A1 + 0.49 g/cm 2 polystyrene + 2 g/cm 2 A1) for 583-MeV/amu iron

beam.



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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

April 1997 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Multilayer Analyses of an Iron Radiation Beam Experiment

6. AUTHOR(S)

J. L. Shinn, J. W. Wilson, E. V. Benton, and F. E Badavi

7. PERFORMING ORGANIZATION NAME(S) ANDADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-0001

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National Aeronautics and Space Administration

Washington, DC 20546-0001

WU 199-45-16-11

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L-17586

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11. SUPPLEMENTARY NOTES

Shinn and Wilson: Langley Research Center, Hampton, VA; Benton: University of San Francisco, San Francisco

CA; Badavi: Christopher Newport University, Newport News, VA.

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13. ABSTRACT (Maximum 200 words)

A nonperturbative analytic solution of Green's function for heavy ion transport in a single medium is extended to

multilayer transport media. This extension is implemented in the Green's function code (GRNTRN), which is

being validated with laboratory experiments. Good agreement is achieved between an iron radiation beam experiment and GRNTRN only when the interactions of the iron radiation beam with several important layers of material

located upstream of the target within the beamline are considered.

14. SUBJECT TERMS

Heavy ion transport; Green's function; Etch track detectors

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