Approximate Green s Function Methods for HZE Transport in Multilayered Materials

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1 NASA Technical Memorandum 459 Approximate Green s Function Methods for HZE Transport in Multilayered Materials John W. Wilson, Francis F. Badavi, Judy L. Shinn, and Robert C. Costen OCTOBER 993

2 NASA Technical Memorandum 459 Approximate Green s Function Methods for HZE Transport in Multilayered Materials John W. Wilson Langley Research Center Hampton, Virginia Francis F. Badavi Christopher Newport University Newport News, Virginia Judy L. Shinn and Robert C. Costen Langley Research Center Hampton, Virginia

3 Abstract A nonperturbative analytic solution of the high charge and energy (HZE) Green's function is used to implement a computer code for laboratory ion beam transport in multilayered materials. The code is established to operate on the Langley nuclear fragmentation model used in engineering applications. Computational procedures are established to generate linear energy transfer (LET) distributions for a specied ion beam and target for comparison with experimental measurements. The code was found to be highly ecient and compared well with the perturbation approximation. Introduction Green's function was identied as the liely means of generating ecient high charge and energy (HZE) shielding codes for space engineering that are capable of being validated in laboratory experiments (ref. ). A derivation of Green's function as a perturbation series gave promise for the development of a laboratory-validated engineering code (ref. 2), but computational ineciency provided a maor obstacle to the code development (ref. 3). More recently, nonperturbative approximations to the HZE Green's function have shown promise in providing an ecient validated engineering code (refs. 4 and 5). Described in the present report is a laboratory code using a nonperturbative Green's function to derive linear energy transfer (LET) spectra behind multilayered targets for ion beams with Z (charges) 28 corresponding to the maor components of the galactic cosmic ray spectrum. Green's Function for a Single Medium We restrict our attention to the multicharged ions (atomic number Z and atomic mass A ) for which the Boltzmann equation may be reduced (ref. 6) es (E)+ (x; E) () where (x; E) is the ion ux at x with energy E (in MeV/amu), S e (E) is the change in E per unit distance, is the total macroscopic reaction cross section, and is the macroscopic cross section for the collision of an ion of type to produce an ion of type. The solution to equation () is to be found subect to the boundary condition (;E)=f (E) (2) which, for laboratory beams, has only one value of for which f (E) is not zero and where f (E) is described by a mean energy E o and energy spread such that f (E) = p exp 2 " # (E E o ) The usual method of solution is to proceed toward solving equation () as a perturbation series (refs. and 6). In practice, the computational requirements limit the usefulness of the technique for deep penetration (ref. 3). (3)

4 The Green's function (Gm) is introduced as a solution S (E)+ Gm(x; E; Eo) = G m (x; E; E o) (4) subect to the boundary condition Gm(;E;Eo)=m (E Eo) (5) where m is Kronecer's and (E Eo) is Dirac's function. The solution to equation () is given by superposition as (x; E) = XZ G x; E; E f E de (6) If G (x; E; E ) is nown as an algebraic quantity, the evaluation of equation (6) may be accomplished by simple integration techniques, and then the associated errors in numerically solving equation () are avoided (ref. 7). The above equations can be simplied by transforming the energy into the residual range (r) as ZE and dening new eld functions as Thus, equation @ r= de =e S E (7) x; r = S e (E) (x; E) (8) Gm x; r;rm = S e(e) Gm ^f x; E; E (9) r = e S (E) f (E) () X Gm x; r;rm = G m x; r ;r m () with the boundary condition Gm ;r ;rm = m r r m (2) and with the solution to the ion elds given by x; r XZ = Gm x; r ;rm ^fm rm drm (3) m Note that, which is the range scale factor as r = mrm, is taen as = Z 2 =A. solution to equation () is written as a perturbation series Gm x; r;r m = X i G (i) m x; r;rm The (4) 2

5 in which and G m () x; r ;r m = g() m x + r rm (5) G m () x; r ;rm m g (; m) (6) x m where G () m (x; r ;r m) is zero unless m r + x r m m r + x (7) for m >.If >m, as can happen in neutron removal, the negative of equation (6) is used and the upper and lower limits of equation (7) are switched. The higher terms are approximated as G m (i) x; r ;rm X ; 2 ;:::;i 2 ; ; im g(; ; 2 ;; i ;m) x m (8) In the above equations, g () =e x (9) and g ( ; 2 ;:::;n; n+ )= g( ; 2 ;:::; n ;n)g( ; 2 ;:::; n ; n+ ) n+ n (2) Note that G (i) m (x; r ;rm) is purely dependent on x for i>, which we represent as G (i) m (x). (See ref. 3.) In terms of the above, the solution to equation () becomes (from ref. 3) x; r = e x ^f r + x + X m;i h G (i) m (x) bf m rm` F b m r i mu (2) In equation (2), rmu and r m` are given by the upper and lower limits of the inequality of equation (7). The symbol F bm(rm) refers to the integral spectrum bfm r Z m = r m ^fm(r) dr (22) We note that with bfm r m Fm E (23) Fm E = Z E fm(e) de (24) 3

6 and r m = Z E de=e S m(e) (25) We now introduce nonperturbative terms for the summation in equation (2). First, we recall that the g function of n arguments was generated by the perturbation solution of the transport equation neglecting ionization energy loss (ref. ) given + g m (x) = g m (x) (26) subect to the boundary condition g m () = m (27) for which the solution is g m (x)= m g (m)+ m g (; m)+ (28) It is also true that g m (x)= X g (x y) g m (y) (29) for any positive values of x and y. Equation (29) may be used to propagate the function g m (x) over the solution space after which Gm h i x; r ;rm e x g m (x) m x + r rm e x m + (3) x m The approximate solution of equation () is then given by x; r e x ^f r + x + X m h i g m (x) e x m h bf m r mu b F m r x m m`i (3) which is a relatively simple quantity (ref. 4). Green's Function in a Shielded Medium The maor simplication in the Green's function method results from the fact that the scaled spectral distribution of secondary ions to a rst approximation depends only on the depth of penetration as seen in equations (6), (8), and (3). Our rst approach to a multilayered Green's function will rely on this observation and assume its validity for multilayered shields. If we consider a domain labeled as \" that is shielded by a second domain labeled as \2," the number of type ions at depth x in due to type m ions incident on domain 2 of thicness y is X g 2m (x; y) = g (x) g 2m (y) (32) 4

7 The leading term in equation (32) is the penetrating primaries as g 2m (x; y) =e x 2 y h m+ g 2m (x; y) e x 2 y i m (33) in which all higher order terms are within the bracets of equation (33). The rst term of the scaled Green's function is then G () 2m x; y; r ;r m = e x 2 y m 2x + r r m 3 y (34) in which is the range scale factor for the two media = R (E) R 2 (E) (35) The ratio of range in water to range in aluminum for proton beams (eq. (35)) is shown in setch A. We tae a single value for corresponding to 6 MeV/amu. The secondary contribution is similarly found by noting that equation (7) becomes m r + x + y rm m r + x + y (36) from which the average spectrum is evaluated. The full approximate Green's function is then G 2m x; y; r ;r m e x 2 y m x + y + r r m + h g 2m (x; y) e x 2 y m (x + y) m i (37) Equation (37) is our rst approximation to the Green's function in a shielded medium (two layers) and is easily modied to multilayers. We now consider the rst spectral modication.. Range of H 2 O Range of Al E, MeV Setch A 5

8 Showing that the rst collision term has the properties 8 G () >< 2m x; y; r ;r m = >: m e m x 2m y m 2m e x 2 y m rm = 9 r mu >= rm = r m` >; (38) is easy; we use these properties to derive a simple correction for the average spectrum as G () g() x; y; 2m r ;r 2m (x; y) m = (x + y) + bm(x; y) rm m r m (39) where g () 2m (x; y) is the rst collision term of equation (39) and r m = r mu + r m` 2 (4) is the midpoint of r m between its limits given by equation (36). The bm term of equation (39) has the property that Z r mu r m` bm(x; y) r r m dr = (4) thus ensuring that the rst term of equation (39) is, indeed, the average spectrum as required. The spectral slope parameter is found to be bm(x; y) = m m e mx 2m y 2m e x 2 y (x + y) m m A similarly simple spectral correction can be made to the higher order terms. The spectral correction given in equation (42) will be included in the present Green's function code. LET Spectra for Laboratory Beams We use the boundary condition appropriate for laboratory beams given by equation (3). The cumulative spectrum is given by F (E) = 2 erf E Eo p 2 (42) (43) The cumulative energy moment needed to evaluate the spectral correction is E (E) = 2 E o erf " E Eo p + p exp (E E o) 2 # (44) The average energy on any subinterval (E;E2) is then E = E (E) E (E2) F (E ) F (E 2) (45) 6

9 The beam-generated ux is x; y; r = e x 2 y ^f + X m;i + X m r + x + y G (i) m (x; y) h bfm r mu b Fm r m` b () m (x; y)2 3h rm E r m bfm rmu Fm b i rm` i (46) where E is evaluated using equation (45) with E and E2 as the lower and upper limits associated with r m` and r mu, respectively. The dierential uence spectra for a 6-MeV/amu 56 Fe beam with a 2.5-MeV/amu standard deviation incident on a water slab are shown in gure. A single layer of 5 cm of water is shown in gure (a) and a double layer of 2.5 cm of water followed by 2.5 cm of water is shown in gure (b). A consistency chec is performed on the multilayered code by comparing it with the single-layered computation when the two layers are of the same size and composition. The ratio of multilayered results to single-layered results diers by less than 6 percent. The LET distribution is found by using the methods of reference 8. The corresponding LET spectra are shown in gure 2. The highest LET pea is due to the primary beam and the iron fragments. The successive peas below iron are due to lower atomic weight fragments. The lowest LET pea consists of relativistic charge fragments of p, d, and t particles that are produced in abundance in HZE collisions (ref. 9). The pea near MeV/cm consists of he- and -particles that are also produced abundantly. Many of the HZE fragments are produced with a charge near the proectile charge, as Shinn, Townsend, and Wilson found earlier for hydrogenic targets (ref. ). Note that no distinguishable dierence exists between the LET spectra of the single-layered code (g. 2(a)) and the multilayered code (g. 2(b)) at the same penetration depth in water, which further demonstrates code consistency. A series of evaluations for a (2.24-g/cm 2 ) lead-scattering foil is shown in gure 3. The lead-scattering foil is usually part of the accelerator beam line (at the Lawrence Bereley Laboratory Bevalac accelerator) with the result that the fragments from the lead target are seen as contamination. Clearly, these fragments must be modeled to properly interpret the attenuation of the beam in the water target in actual experiments. The corresponding LET spectra at various depths of a water target are shown in gure 4. The importance of the fragmentation in the scattering foil is seen in comparing gure 4(a) with gure 4(b). Note that fewer of the fragmentations result in fragments near the beam charge for the lead foil in comparison with the water target, as seen by comparing the relative magnitudes of the three highest LET peas in gures 2(a) and 4(a). These dierences are part of the reason why hydrogenic targets are important to space radiation protection. At greater depths the LET distributions begin to overlap, and distinguishing the dierent charge groups becomes more dicult. Such LET spectra will be compared with experimental measurements in the near future. Concluding Remars A formalism for the evaluation of Green's function in multilayered target congurations has been derived and a computer code generated. The code satises a consistency chec for multilayered-material calculations when the layers are of uniform composition by comparing the results with the single-layered code. The importance of the multilayered code in the analysis of 7

10 experimental ion beams has been shown by demonstrating the eects of a lead-scattering foil in the Bevalac beam line. An analysis of such experiments is in progress. NASA Langley Research Center Hampton, VA September, 993 References. Wilson, J. W.; Lamin, Stanley L.; Farhat, Hamidullah; Ganapol, Barry D.; and Townsend, Lawrence W.: A Hierarchy of Transport Approximations for High Energy Heavy (HZE) Ions. NASA TM-48, Wilson, John W.; Townsend, L. W.; Lamin, S. L.; and Ganapol, B. D.: A Closed-Form Solution to HZE Propagation. Radiat. Res., vol. 22, 99, pp. 223{ Wilson, J. W.; and Badavi, F. F.: New Directions in Heavy Ion Shielding. Proceedings of the Topical Meeting on New Horizons in Radiation Protection and Shielding, American Nucl. Soc., Inc., 992, pp. 25{2. 4. Wilson, John W.; Costen, RobertC.; Shinn, Judy L.; and Badavi, Francis F.: Green's Function Methods in Heavy Ion Shielding. NASA TP-33, Wilson, John W.; Badavi, Francis F.; Costen, Robert C.; and Shinn, Judy L.: Nonperturbative Methods for HZE Transport. NASA TP-3363, Wilson, John W.: Analysis of the Theory of High-Energy Ion Transport. NASA TN D-838, 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-257, Wilson, John W.; and Badavi, Francis F.: A Study of the Generation of Linear Energy Transfer Spectra for Space Radiations. NASA TM-44, Wilson, J. W.; Chun, S. Y.; Badavi, F. F.; and John, S.: Coulomb Eects in Low-Energy Nuclear Fragmentation. NASA TP-3352, Shinn, Judy L.; Townsend, Lawrence W.; and Wilson, John W.: Galactic Cosmic Ray Radiation Levels in Spacecraft on Interplanetary Missions. Boo of Abstracts The World Space Congress, 43rd Congress of the International Astronautical Federation and 29th Plenary Meeting of the Committee on Space Research, International Astronautical Federation and Committee on Space Research, Aug.{Sept. 992, pp. 567{568. 8

11 (a) cm of H 2 O followed by 5 cm of H 2 O. (b) 2.5 cm of H 2 O followed by 2.5 cm of H 2 O. Figure. Dierential uence spectra for 6-MeV/amu 56 Fe beam with 2.5-MeV/amu standard deviation incident on water slab. (a) cmofh 2 O followed by 5 cm of H 2 O. (b) 2.5 cm of H 2 O followed by 2.5 cm of H 2 O. Figure 2. LET distribution for 6-MeV/amu 56 Fe beam with 2.5-MeV/amu standard deviation incident on water slab. (a) cmofh 2 O. (b) 5cmofH 2 O. Figure 3. Dierential uence for 525-MeV/amu 56 Fe beam with 2.5-MeV/amu standard deviation after passing through 2.24-g/cm 2 lead-scattering foil and water target. (c) cm of H 2 O. Figure 3. Concluded. (a) cmofh 2 O. (b) 5cmofH 2 O. Figure 4. LET distribution for 525-MeV/amu 56 Fe beam with 2.5-MeV/amu standard deviation after passing through 2.24-g/cm 2 lead-scattering foil and water target. (c) cm of H 2 O. Figure 4. Concluded.

12 REPORT DOCUMENTATION PAGE Form Approved OMB No Public reporting burden for this collection of information is estimated to average hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 25 Jeerson Davis Highway, Suite 24, Arlington, VA , and to the Oce of Management and Budget, Paperwor Reduction Proect (74-88), Washington, DC AGENCY USE ONLY(Leave blan) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED October 993 Technical Memorandum 4. TITLE AND SUBTITLE Approximate Green's Function Methods for HZE Transport in Multilayered Materials 6. AUTHOR(S) John W. Wilson, Francis F. Badavi, Judy L. Shinn, and Robert C. Costen 5. FUNDING NUMBERS WU PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langley Research Center Hampton, VA PERFORMING ORGANIZATION REPORT NUMBER L SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aeronautics and Space Administration Washington, DC SPONSORING/MONITORING AGENCY REPORT NUMBER NASA TM-459. SUPPLEMENTARY NOTES Wilson, Shinn, and Costen: Langley Research Center, Hampton, VA; Badavi: Christopher Newport University, Newport News, VA. 2a. DISTRIBUTION/AVAILABILITY STATEMENT 2b. DISTRIBUTION CODE Unclassied{Unlimited Subect Category ABSTRACT (Maximum 2 words) A nonperturbative analytic solution of the high charge and energy (HZE) Green's function is used to implement a computer code for laboratory ion beam transport in multilayered materials. The code is established to operate on the Langley nuclear fragmentation model used in engineering applications. Computational procedures are established to generate linear energy transfer (LET) distributions for a specied ion beam and target for comparison with experimental measurements. The code was found to be highly ecient and compared well with the perturbation approximation. 4. SUBJECT TERMS 5. NUMBER OF PAGES Radiation shielding; Multilayer; Green's function 5 6. PRICE CODE A3 7. SECURITY CLASSIFICATION 8. SECURITY CLASSIFICATION 9. SECURITY CLASSIFICATION 2. LIMITATION OF REPORT OF THIS PAGE OF ABSTRACT OF ABSTRACT Unclassied Unclassied NSN Standard Form 298(Rev. 2-89) Prescribed by ANSI Std. Z