An analytic solution for high energy heavy ion transport assuming straightahead and velocity conserving interactions with constant nuclear cross reactions is given in terms of a Green's function. The series solution for the Green's function is rapidly convergent for most practical applications. The Green's function technique can be applied with equal success to laboratory beams as well as to galactic cosmic rays allowing laboratory validation of the resultant space shielding code

Ganapol, B. D.

Lamkin, S. L.

Townsend, L. W.

Wilson, John W.

English

NASA Technical Reports Server

A CLOSED FORM SOLUTION TO HZE PROPAGATION
John W. Wilson and L. W. Townsend
NASA Langley Research Center
Hampton, VA 23665-5225
S. L. Lamkin
Planning Research Corporation
Hampton, VA 23665-5225
and
B. D. Ganapol
Nuclear and Energy Engineering
University of Arizona
Tucson, AZ 85711
7 .i
C_
i_ ¸ )
GOABSTRACT
An analytic solution for high-energy heavy ion transport assuming
straightahead and velocity conserving interactions with constant nuclear cross
reactions is given in terms of a Green's function. The series solution for the Green's
function is rapidly convergent for most practical applications. The Green's function
technique cart be applied with equal success to laboratory beams as well as to galactic
cosmic rays allowing laboratory validation of the resultant space shielding code.
Ont.] _s
https://ntrs.nasa.gov/search.jsp?R=19900017324 2020-03-19T21:19:33+00:00Z
Introduction
2
Several approaches to the solution of high energy heavy ion propagation
have been developed (1-12) over the last 20 years. All but one (5) have assumed the
straightahead approximation and velocity conserving interactions. Only two
(5, 8) have incorporated energy-dependent nuclear cross sections. The approach by
Curtis et al. (I) for a primary ion beam represented the first generation secondary
fragments as a quadrature over the collision density of the primary beam. Allkofer
and Heinrich (2) used an energy multigroup method in which an energy-
independent fragmentation transport approximation was applied within each
energy group after which the energy group boundaries were moved according to
continuous slowing down theory (-dxdE). Chatterjee et al. (3) solved the energy-
independent fragment transport equation with primary collision density as a source
and neglecting higher-order fragmentation. The primary source term extended only
to the primary ion range from the boundary. The energy-independent transport
solution was modified to account for the finite range of the secondary fragment ions.
Wilson (4) derived an expression for the ion transport problem to first order (first-
collision term) and gave an analytic solution for the depth-dose relation. In ref. 5,
the more common approximations used in solving the heavy ion transport
problem were examined. The effects of conservation of velocity on fragmentation
3and the straightahead approximation are found to be negligible for cosmic ray
applications. Solution methods for representing the energ'y-dependent nuclear cross
sections are developed. Letaw et al. (6) approximate the energy loss term and ion
spectra by simple forms for which energy derivatives are evaluated explidtly (even
if approximately). The resulting ordinary differential equations in position are
solved analytically similar to the method of Allkofer and Heinrich (2). This
approximation results in a decoupling of motion in space and change in energy. In
Letaw's formalism, the energy shift is replaced by an effective attenuation factor.
Wilson adds the next higher-order (second collision) term (7). This term was found
to be very important in describing 670 MeV/amu 2°Ne beams. The three-term
expansion of ref. 7 was modified to include the effects of energy variation of the
nuclear cross sections (8). The integral form of the transport equation (5) was
further used to derive a numerical marching procedure to solve the cosmic ray
transport problem (9). This method can easily include the energy-dependent
nuclear cross sections within the numerical procedure. Comparison of the
numerical procedure (9) with an analytical solution to a simplified problem (10)
validates the solution technique to about 1 percent accuracy. Several solution
techniques and analytic methods have been developed for testing future numerical
solutions to the transport equation (11, 12). More recently, an analytic solution for
the laboratory ion beam transport problem has been derived assuming a
straightahead approximation, velocity conservation at the interaction site, and
energy-independent nuclear cross sections (13).
In the above overview of past developments, the applications split into two
separate categories according to a single ion species with a single energy at the
boundary versus a broad host of elemental types with a broad continuous energy
spectrum. Techniques requiring a representation of the spectrum over an array of
energy values require vast computer storage and computation speed for the
laboratory beam problem to maintain sufficient energy resolution. On the other
hand, analytic methods (5, 9) are probably best applied in a marching procedure (9)
which again has within it a similar energy resolution problem. This is a serious
limitation since we require a final HZE code for cosmic ray shielding which has been
validated by laboratory experiments. In the present paper, we examine new
methods which appear to overcome these difficulties.
The HZE Transport Problem
In moving through extended matter, heavy ions lose energy through
interaction with atomic electrons along their trajectories. On occasion, they interact
violently with nuclei of the matter producing ion fragments moving in the forward
direction and low energy fragments of the struck target nucleus. The transport
5equations for the short-range target fragments can be solved in closed form in terms
of collision density (ref. 5). Hence, the projectile fragment transport is the
interesting unsolved problem. In previous work, the projectile ion fragments were
treated as if all went straightforward. We continue with this assumption herein,
noting that an extension of the beam fragmentation model to three dimensions is
being developed (ref. 14).
With the straightahead approximation and the target secondary fragments
neglected (ref. 5), the transport equation may be written as
a S'_(E)+crj _j(x,E)= _ mjkakq_k(x,E)
aE k>j
where _i (x, E) is the flux of ions of type j with atomic massAj at x moving along the
x-axis at energy E in units of MeV/amu, o i is the corresponding macroscopic nuclear
absorption cross section, Sj (E) is the change in E per unit distance, and mjk is the
fragmentation parameter for ion j produced in collision by ion k. The range of the
ion is given as
E !
Ritz) Jo
The solution to equation (1) is to be found subject to boundary specification at x = 0
(2)
and arbitrary E as
6Cj(0, E) = Fj (E) (3)
Usually Fj (E) is taken as an incident laboratory ion beam spectrum or the cosmic ray
spectrum.
It follows from Bethe's theory that
AjZ_2
for which
_Zp 
The subscript p refers to proton. Equation (5) is quite accurate at high energy and
(4)
(5)
only approximately true at low energy because of electron capture by the ion which
effectively reduces its charge, higher-order Born corrections to Bethe's theory, and
nuclear stopping at the lowest energies. Herein, the parameter vi is defined as
Vj= Zj2/Aj (6)
so that
vjRj(E)= vk R k CE) (7)
Equations (6)and (7)are used in the subsequent development, and the energy
variationin vjisneglected. The inversefunctionof P_ (E)isdefined as
E=Rj'[RjcE)] (s)
and plays a fundamental role subsequently.
7Impulse Response
One form of Green's function is the impulse response corresponding to a 5-
like source term at the boundary. We therefore seek a solution of
O s__) + oj OjM(x,E, E')= Y_mj__kC_ (x,E;Z')
3E k
where the boundary condition is
G_M(0, E; Z') = _jM_ _" E') (10)
for which transport solution may be written as
0j(x,E)=_ Jo GJM (x'E; E') FM (E') dE' (11)
M
The solution to equations (9) and (10) is straightforward, even if tedious (4, 7, 13) and
is arrived at using the method of characteristics (15). The solution is expressed as a
series as
GjM (x, E; E') = _ r,(i) (x, E; E')
"'jM
i
where
G(O) (x, E; E')jM
r
=__t__exp(-ojx)5jM5[
sj (E) x + Rj (E)- R M (E') ]
(12)
(13)
G(:) (x, E; E')jM =_l_mj McrM vj
8
1 }
so long as
V-M-M[RM(E')'x]<Rj('E)<_; M.R (E')'xvj
where
(14)
(15)
V M + Vj x]- 2VM RM(E') - [Rj(E)+11' VM-Vj VM Vj
(16)
Otherwise _.(1) (x, E; E') is zero. After a complicated but straightforward
"-'jM
manipulation, a similar result may be obtained r': (2) (x, E; E') as
"jM
G(2) Z cr;_ O'kM vj V -(:rM XM1-O"k Xld -O'j XjljM (x, E; E') = ........ e -ek (E)IvM-vkl L
-o"M XMu -Ork x_ -Oj xju
(17)
where XMu, Xku, XMp and xkn are values of X M and x k evaluated at the corresponding
upper and lower limits of xj and
XM --[VM RM (E')-Vk (Rk (E)+ x) + (Vk-Vj)Xj]/ (VM -Vk)
Xk :[VM (RM (E)+ x)- VM RM (E')" (VM- Vj) Xj]/(VM - Vk)
(18)
(19)
The requirements that xM and xk be bounded by the interval 0 to x-xj yields
0
vk[ Rk _) + x ]- VMRM0_')
Vk - Vj
t<xj<
/
x
VM[ R M (E) + x ]- v M R M (E')
v M -vj
(20)
as the appropriate limiting values in equation (17) when v M > v k > v i . In the above
brace, we always choose the most restrictive value for the limit. The requirement of
equation (20) also implies the result that
_ {_M__-x}____<, {."_' ,,_c_'_-x/./ (21)
as the range over which the result of equation (17) is not zero. In the event that
Vk > VM > Vj, then
/
0
VM[RMrE) + x ]- VMRMrE')
VM - Vj
<x_<
As a result of equation (22)
} { ,}R_I[VMRMOE')-x <E<R_ RMOE')-_MX|Vk
x
Vk[ R k (E) + x]- v M R M (E')
vk - vj
(22)
(23)
In the event that v M > vj > v k, it follows that
10
I X
0<xj_< VM [ RM (E) + x _ RM ('E')] / (VM. Vj)
[vMR__E')- vkR_CE)-vkx]/%- v_)
(24)
where the lesser of the three values in the brace is used as the upper limit of xj for
which G (2) of equation (17) is not zero. As a result of equation (24)
R_[ RMCE') x]<E<D't[VMRM('E')-x]
" - -- LXk t Vk
(25)
Note that
r (Vk v j)
(VM- Vk)t.
](vM- v j)
oM - /
(VM - _k ) (3"k3
(26)
Higher-order terms are similarly derived. Approximate expressions have been
obtained as
G(n) (x, E; E') =jM
k. h,.-. h.2
where
(Yjk(YkjI ... (Yjn-2 M
Ejl = Wk
g (j,k,jl...,jn.2.M) /CEju- E9)
V M > V k > Vj
V k > V M > Vj
V M > Vj > V k
(27)
(28)
and
I
V M > V k > Vj
V k > V M > Vj
V M > Vj > V k
(29)
and the g-functions of n + 1 arguments are defined as
g (Jl) = exp (- rrj, x)
g (Jl,j2,..-j,,j,+l) =g (jl,j2...,jn-l.j_)-g (Jl, j2"",ja-t,j,+l)
(_j.., - Oj.
11
(30)
(31)
The expression for c, (n) given by equation (27) is taken as zero unless
"-'jM
Ejl _<E < Eju (32)
Portions of the Green's function are shown for incident 2°Ne beams at
E' = 600 MeV/amu at x = 20 cm in figures 1 to 3. The contribution from G a) is
shown as the solid curve, G _1) + G (2) is shown as the dash-dot curve, and
G¢1) + G _2)+ G _3) is shown as the dashed curve. The dash-double-dot curve
representing the inclusion of G (4) in the sum can hardly be distinguished signifying
convergence to a high degree of accuracy. A fuller presentation of the Green's
function for I60 fragments is _ven in figure 4, and a presentation of the _2C Green's
function is given in figure 5. From the present result, the solution for any arbitrary
boundary condition may be found using equation (11).
Although the present formalism presents a closed form solution for the more
common form of the HZE propagation problem, many tasks remain before the HZE
propagation problem is adequately solved. The inclusion of energy-dependent
nuclear cross sections is known to be very important in obtaining accurate solutions
12
to some problems (16). Treating the momentum spread of the fragments is more
complicated for the higher-order terms. The inclusion of the light fragment spectra
is a difficult challenge (I7). Finally, the three-dimensional aspects of the problem
have only partially been treated (I4). Even these shortcomings of the HZE
propagation problem remain without the mention of uncertainties in nuclear cross
sections (18) or atomic/molecular cross sections. Clearly much work remains.
References
13
.
.
.
.
.
.
S. B. Curtis, W. R. Doherty, and M. C. Wilkinson, Study of Radiation
Hazards to Man on Extended Near Earth Missions. NASA CR-1469,
Washington, D.C., 1969.
O. C. Allkofer and W. Heinrich, Attenuation of Cosmic Ray Heavy Nuclei
Fluxes in the Upper Atmosphere by Fragmentation. Nuclear Physics B71,
429 (1974).
A. Chatterjee, C. A. Tobias, and J. T. Lyman, Nuclear Fragmentation in
Theraputic and Diagnostic Studies with Heavy Ions. In Spallation Nuclear
Reaction and Their Applications (B.S.P. Shen and M. Merker, Eds.) Reidel,
Dordrecht/Boston, 1977.
J'. W. Wilson, Depth dose relations for heavy ion beams. Virginia Journal of
Science _ 136 (1977).
5. W. Wilson, Analysis of the Theory of High-Energy Ion Transport. NASA
TND-8381, Washington, D.C., 1977.
J. Letaw, C. H. Tsao, and R. Silberberg, Matrix Methods of Cosmic Ray
Propagation. In Composition and Origins of Cosmic Rays (M. M. Shapiro,
Ed.), Reidel, Dordrecht/Boston, 1983.
oo
o
10.
11.
12.
13.
14
]'. W. Wilson, Heavy Ion Transport in the Straightahead Approximation.
NASA TP-2178, Washington, D.C., 1983.
J. W. Wilson, L. W. Townsend, H. B. Bidasaria, W. Schimmerling,
M. Wong, and ]'. Howard, 2°Ne Depth-dose Relations in Water. Health
Physics 46, 1101 (1984).
J. W. Wilson, and F. F. Badavi, Methods in Galactic Heavy Ion Transport.
Radiation Research 108, 231 (1986).
J. W. Wilson and L. W. Townsend, A Benchmark for Galactic Cosmic-Ray
Transport Codes. Radiation Research _ 201 (1988).
B. D. Ganapol, L. W. Townsend, and J'. W. Wilson, Benchmark Solutions for
the Galactic Ion Transport Equations: Energy and Spatially Dependent
Problems. NASA TP-2878, Washington, D.C., 1989.
L. W. Townsend, B. D. Ganapol, and J'. W. Wilson, Benchmark Solutions for
Heavy Ion Transport Code Validation. 37th Annual Meeting of the
Radiation Research Society, Seattle, WA, March 18-23, 1989.
J. W. Wilson, S. L. Lamkin, H. Farhat, B. D. Ganapol, and L. W. Townsend,
A Hierarchy of HZE Transport Approximations. NASA TM- ___, 1989.
14.
15.
16.
17.
18.
15
W. Schimmerling, J'. Howard, M. Rapkin, and M. Wong, The Propagation of
Relativistic Heavy Ions in Multielement Beam Lines. Medical Physics 13,
217 (I986).
J. W. Wilson and S. L. Lamkin, Perturbation Theory in Charged Particle
Transport. Nuclear Science and Engineering 57, 292 (1975).
L. W. Townsend and J'. W. Wilson, An Evaluation of Energy-Independent
Heavy Ion Transport Coefficient Approximations. Health Physics _ 409
(1988).
J. W. Wilson, L. W. Townsend, J. E. Nealy, S. Y. Chun, B. S. Hong, W. W.
Buck, S. L. Lamkin, B. D. Ganapol, F. Khan, and F. Cucinotta, BRYNTRN: A
Baryon Transport Code. NASA TP-2887, Washington, D.C., 1989.
L. W. Townsend and J. W. Wilson, Nuclear cross sections for hadronic
transport. Trans. Am. Nud. Soc. 5_fi,6271 (1988).
16
FIGURE CAPTIONS
Fig. 1- Sequenceof approximations of the I70 flux spectrum after 20 cm of water
for first order (-), second order (-. -), third order (---), and fourth order (.... )
theories.
Fig. 2- Sequence of approximations of the 160 flux spectrum after 20 cm of water
for first order (-), second order (-. -), third order (-), and fourth order (.... )
theories.
Fig. 3 - Sequence of approximations of the 12C flux spectrum after 20 cm of water
for first order (-), second order (-. -), third order (--), and fourth order (.... )
theories.
Fig. 4- Green's function for 160 flUX spectrum response to a 600 MeV/amu 2°Ne
flux at the boundary.
Fig. 5 - Green's function for 12C flux spectrum response to a 600 MeV/amu 2°Ne
flux at the boundary.
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A closed form solution to HZE propagation

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