Hampton,. Virginia. National Aeronautics and. Space Administration. Office of ...... Mark. R.: Heavy-Ion. Transport. Code. Calculations and. Comparison. With.
NASA
Technical
A Hierarchy
Memorandum
of Transport
Approximations Energy
John
W.
Langley
Heavy
PRC
(HZE)
Ions
Center
University Tucson,
Virginia
Lawrence
L. Lamkin Kentron,
Inc.
Aerospace
Technologies
Hampton,
Virginia
Hamidullah
Farhat
Hampton
University
Hampton,
Virginia
National Aeronautics and Space Administration Office of Management Scientific and Technical Information Division 1989
High
Barry
Research
Stanley
for
Wilson
Hampton,
4118
Langley Division
Hampton,
D.
Ganapol of Arizona Arizona W.
Townsend
Research Virginia
Center
-TI IT
Contents 1. Introduction
................................
2. Energy-Independent 2.1.
Neon
2.2.
Iron
Beam Beam
3. Monoenergetic 3.1.
Total
3.2.
Monoenergetic
4. Realistic
Flux
Ion
5. Approximate
Flux
..........................
1
..........................
1
Transport
..........................
2
Ion
..........................
5
Transport
Beams
Comparisons Beam
Beams
........................ Results
Solutiors
5.1.
Approximate
Monoenergetic
5.2.
Approximate
Realistic
References
10
......................
11
............................
Spectral
6. Recommended
1
Methods
Beams
16 ...................... Beams
22
...................
......................
23
..........................
29
.................................
PRECEDING
22
33
PAGE BLANK
NOT
RLM1ED
_111
Abstract The terials being equate
transport is studied
neglected. for
less than
most
energy
energy
A three-term practical beams
approximate
and
formalism
PRECEDING
heavy
(HZE)
dependence
of the
perturbation
applications
30 g/cm 2 of material.
monoenergetic An
of high with
for
which
The
differential
realistic
ion
beam
to estimate
PAGE
BLANK
v
NOT
through
nuclear
expansion
for
is given
ions
bulk
cross
appears
penetration energy
flux
spectral higher
ma-
sections to be ad-
depths
are
is found
for
distributions. order
RLMED
terms.
W
-1 1 Ii
1. Introduction Although heavy ion transport codes for use in space applications are in a relatively advanced stage at Langley Research Center (refs. 1 through 3), it seems prudent to further develop the theory for comparison with laboratory experiments, which has been recently neglected since initial efforts began several years ago (refs. 4 through 7). In the present report, we begin with the most simplified assumptions for which the problem may be solved completely. Solutions to a more complete theory may then be compared with prior results as limiting cases. In this way, the more complete but approximate analysis will have some basis for evaluating the accuracy of the solution method. The lowest order approximation will be totally energy-independent. The most complicated solution to be considered herein will have energy-independent nuclear cross sections but will treat the energy-dependent atomic/molecular processes and the energy spread of the primary beam. A fully energy-dependent theory must await further development, although some terms have been previously evaluated (ref. 6). 2. Energy-Independent
Flux
If the ion beam is of sufficiently high energy that the energy shift due to atomic/molecular collisions brings none of the particles to rest in the region of interest, then
[o+ ]Cj(x) =
mjk k Ck(x)
(2.1)
k
where Cj(x) is the flux of type j ions, aj is the nuclear absorption cross section, the fragmentation parameter for producing type j ions from type k. The solution incident ion type J is given in terms of a set of g-functions as follows: g(Jl) g(Jl,J2,... for which the solution
,Jn,Jn+l)
= exp(-crjlx)
(2.2)
. ,Jn-l,Jn) - g(Jl,J2,... O'jn+ l -- O'jn for the type j ion flux is written as
= g(Jl,J2,..
,Jn-l,Jn+l)
¢_°)(x) = 6jj g(j)
¢_l)(x)
= my jag
@ 2)(x) = Z k
and rnjk is for a given
(2.3)
(2.4)
g(j, J) : rnjjaj
mJ kak mkjaJ
¢_3)(x) = _ mjkak k,I
exp(-6jx)
o'j - aj
g(j,k,J)
mklalmljaj
- exp(-6jx)
g(j,k,l,J)
(2.5) (2.6)
(2.7)
with
(2.8) This solution is equivalent to that derived some applications of the above formalism. (ref. 9). 2.1.
Neon Beam
We first contributing
by Ganapol et al. in reference 8. We now consider The cross-section data base is discussed elsewhere
Transport
note in the case of 2ONe incident on water that 19Ne and 19F have only one term in equation (2.8). These are shown in figure 2.1: Also shown in figure 2.1
arethe fluxesof variousisotopesof secondary ion fragments.The effectof successive termsof equation(2.8)is shownin table2.1for the 150flux. It is clear from the table that the fourth and higher order collision terms are completely negligible and that third collision terms are a rather minor contribution. Hence, a three-term expansion as we have used in the past _refs. 5, 6, and 7) appears justified. The relative magnitude of the terms contributing to the "Li flux generated by the 2UNe beam is presented in table 2.2. The fourth collision term is negligible at small penetration distances and small, but not negligible, at distances greater than 30 cm. The greater penetrating power of the lighter mass fragments is demonstrated in figure 2.2. Also note the difference in solution character due to the importance of the higher order term. 2.2.
Iron Beam
Transport
We first note in the case of 56Fe incident on water that 55Fe and 55Mn have only one contributing term in equation (2.8). The 54Mn has but two terms, and the slight difference in solution character can be seen in figure 2.3. Results for 52V are also shown. The convergence rate of equation (2.8) is demonstrated in-table 2.3. Again we see the fourth collision term to be negligible, whereas the three-term expansion we have used before seems quite accurate at these depths for these ions. In distinction to prior results, the 160 flux has significant contributions from higher order terms for depths beyond 20 cm as seen in table 2.4. Clearly, a more complete theory using higher order terms is required than previously used for ion beams of particles heavier than 2°Ne. The different solution character of the lighter mass fragments is clearly demonstrated in figure 2.4. Table
2.1.
Normalized Collision
Contributions Terms
for
to i50
20Ne
150
Flux
Transport
flux
From
Successive
in Water
at x o_
Fragment term ¢(I)
10 cm
20 cm
30 cm
40 cm
1.00E0
¢(2)
1.01 E- 1
2.01E-
¢(3)
2.63E-3
1.05E-2
2.36E-2
4.18E-2
6.52E-2
¢(4)
3.31E-5
2.52E-4
8.58E-4
2.03E-3
3.95E-3
Table
2.2.
Normalized Collision
1.00E0
50 cm
1.00E0
1
3.02E-
Contributions Terms
1.00E0 1
4.03E-
to 7Li Flux
for 20Ne
Transport
7Li flux
From
1.00E0 1
5.04E-
Successive
in Water
at x o_
Fragment term
10 cm
20 cm
30 cm
40 cm
50 cm
¢(1)
1.00E0
1.00E0
1.00E0
1.00E0
¢(2)
1.62E-1
3.20E-1
4.72F_,-1
6.18E-1
7.58E-1
¢(3)
1.15E-2
4.53E-2
9.98E-2
1.73E-I
2.63E-I
¢(4)
4.02E-4
3.16E-3
1.04E-2
2.39E-2
4.53E-2
y[[-
1.00E0
1
Table 2.3. Normalized Collision
Contributions
to 52V Flux From Successive
Terms for 56Fe Transport
in Water
52V flux at x o_ Fragment 10 cm
term
20 cm
30
50 cm
40 cm
cm
¢(1)
1.00E0
1.00E0
1.00E0
1.00E0
1.00E0
¢(2)
7.91E-2
1.52E-1
2.37E-I
3.15E-i
3.94E-1
_(3)
2.37E-3
9.48E-3
2.13E-2
3.79E-2
5.91E-2
4,(4)
2.24E-5
1.73E-4
5.93E-4
1.41E-3
2.75E-3
Table 2.4. Normalized Collision
Contributions
to
160
Flux From Successive
Terms for 56Fe Transport
in Water
160 flux at x of Fragment term
10 cm
20 cm
40 cm
50 cm
¢(1)
1.00E0
1.00E0
1.00E0
30 cm
1.00E0
1.00E0
¢(2)
5.87E-1
1.12E0
1.59E0
2.00E0
2.36E0
¢(3)
1.86E- 1
7.08E- 1
1.49E0
2.46E0
3.56E0
3.06E-2
2.63E- 1
9.44E- 1
2.33E0
4.72E0
.O3
.O2 E ? {D t-" O
LL
.01
I 10
0
t 20
I 30
[ 40
I 50
I 60
x, cm
Figure
2.1.
incident
ion
fragment
flux
of
various
isotopes
as
a function
of
depth
in
water
for
a 2°Ne
beam.
3
.02 -
_12 C 7Li /--14 N
/_ .01 .o_
I 10
0
Figure 2.2. beam.
Flux
of light
.03
ion
fragments
I 20
....I 30 x, cm
as a function
1 40
of depth
I 50
in water
I 60
for a 2°Ne
incident
-
.O2 E ?
,/_55
¢O
Fe 52V 54Mn
.01
I 10
0
I 20
I 30 X,
Figure 2.3. incident
Ion fragment beam.
flux
of various
isotopes
I 40
_i 50
I 60
of depth
in water
cm
as a function
4
VIF
for a 56Fe
.03
F
.O2 E ? _D C
.9 x_ tl_
.01
0
Figure
2.4.
Flux
of light
3. Monoenergetic
1 10
ion fragments
I 20
I 30 x, cm
as a function
I 40
of depth
I 60
I 50
in water
for a 56Fe incident
beam.
Ion Beams
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 and produce ion fragments moving in the forward direction and low energy fragments of the struck target nucleus. The transport equations for the short range target fragments can be solved in closed form in terms of collision density (refs. 5 and 6). Hence, the projectile fragment transport is the interesting unsolved problem. In previous work, the projectile ion fragments were treated as if all went straightforward (ref. 4). We continue with this assumption herein, noting that an extension of the beam fragmentation model to three dimensions is being developed (ref. 9). With the straightahead approximation and the target secondary fragments neglected (refs. 4, 5, and 6), the transport equation may be written as
(3.1) k>j where
energy
E) is the flux of ions of type j with atomic E in units of MeV/amu, aj is the corresponding
Cj(x,
mass Aj at x moving along the x-axis at macroscopic nuclear absorption cross
section, Sj(E) is the change in E per unit distance, and for ion j produced in collision by ion k. The range of the
Rj (E)
=
I']_E Jo
mjk is the fragmentation ion is given as
parameter
dE I
The stopping powers used herein are based on Ziegler's fits to a large through 16). There is some controversy as to the stopping powers to be analysis in reference 10 was biased to the stopping power used in the data same as that used in the PROPAGATE code. The values in the HZESEC similar to those in PROPAGATE. Neither the PROPAGATE nor HZESEC
(3.2)
data base (refs. 11 used (ref. 10). The analysis and is the computer code are stopping powers 5
have been compared with the data base collected by Ziegler as far as is known to us. We continue to use Ziegler's work until more definitive comparisons compel us to do otherwise. The solution to equation (3.1) is to be found subject to boundary specification at x = 0 and arbitrary E as Cj(0, E) = Fj(E) Usually Fj(E) is called the incident beam It follows from Bethe's theory that
Sj(E)
(3.3)
spectrum.
= ApZ2
Sp(E)
(3.4)
for which Z2 Rj(E)=
Z2p
(3.5)
The subscript p refers to proton. Equation (3.5) is quite accurate at high energy and 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 vj is defined as
=
(3.6)
so that vj Rj(E)
vk Rk(E)
(3.7)
Equations (3.6) and (3.7) are used in the subsequent development, vj is neglected. The inverse function of Rj (E) is defined as
and the energy variation
E = Ry 1 [Rj(E)]
(3.8)
and plays a fundamental role subsequently. For the purpose the coordinate transformation (refs. 5 and 6),
of solving
yj =- x - Rj(E)
x + R (E)
in
equation
(3.1), define
(3.9)
J
and new functions Xj(yj,_j)
= Sj(E)
Cj(x,E)
(3.10)
J where
(3.11)
for which equation
(3.1) becomes
(2°)
k
(3.12)
m 3k_k Vk
6
'l tli
where the crj is assumed to be energy integration with the integrating factor,
1 = exp[_rj(_j
#j(r/j,_j) results
Solving
independent.
equation
(3.12)
by using
+ r/j)]
line
(3.13)
in
X.j(r/j,_j)
1 exp[-_aj(_j
=
+ r/j)]
Xj(-_j,_j)
+ 2 j__ exp _°i(r/-
r/J) _k
m_k_k_
(3.14)
Xk(nk,_k) dr/'
where ,
r/k= 2_ r vk-vjr/r and the boundary
Consider
condition
a Neumann
and the second
term
x_l)(r/j,,j)
An expression
_J
. _v k+vj
(eq. (3.3)) is written
series for equation
x_O)(r/j,_j)
.k - u3
uk + uj r/, +
)
[
(3.15)
as
(3.14) for which the first term is
=exp[-laj(r/j
+ _j)]
Sj[RSI(_j)
] Fj[Rfl(_j)]
is
=1
frlj exp [laj(r/,2 J_(j
for X!.2)(r/j, (j) is derived
r/j)] _
k
mjkak_exp[
once equation
-x_ak(r/k' --'k)]'
(3.17) is reduced
and higher
can be found by continued iteration of equation (3.14). These expressions (3.17)) are now simplified for a monoenergetic beam of type M ions. The boundary condition is now taken as Fj(E) where Thus,
6jM
is
(3.16)
the Kronecker
delta,
=
_jM
order
delta,
and
(3.18)
6(E-Eo)
6( ) is the Dirac
terms
(eqs. (3.16)
and Eo is the incident
beam
energy.
for which X_.O) becomes
X_°)(r/j,_j)
1 = 6jMexp[-_aj(r/j
+ _j)]
6[_j-Rj(Eo)]
(3.20) 7
and
X_ I) becomes
= lnj _ f-¢j
x_l)(llj,_j)
mjM°'M-_M vj
exp [1_o'j77j
× 6[_M--RM(Eo)] where _/ is given occurs at
by equation
(3.15)
dr
for k = M.
(3.21) The
tit = VM21/M-uj RM(E°) provided
that
,1' lies on the
, _J)=
x_l)(rtJ The simplified form equation (3.14):
interval
mjMaMVj I_M ---_jl
in equation
x_?)(_;, G) =
-_j
(3.23)
21 Z k
1 I + _tM) ] -- -_O'M(_]M
< r/
v k > vj. In the braces most restrictive value for the limit. The requirement that
of equation
) -4-x] - t_M RM(Eo
of equation
(3.36)
v M -/2 k
x M and
over which
) "4-x] -4-(v k -- vj)xj
) + x] -- tJM RM(Eo
as the appropriate limits for the integral in equation (3.38), we always choose the of equation (3.38) also implies the result
as the range then
(3.35)
)
v M -- tJk VM[RM(E
that
- u k Rk(E
as
Xk
requirement
(3.34)
= u M RM(Eo)
XM
The
as
(3.33)
uM RM(E°)-vjx] Vk
is not zero.
} { < xj
v M > vj,
)
/
(3.40)
(3.40)
(3.41) 9
In the eventthat vM
> vj > P'k, it follows
that x
v M [RM(E)
+ x - RM(Eo)] (3.42)
VM -- vj VMRM(Eo)
-- v k Rk(E)
-
vkx
pj - v k where
the
integral
lesser
of the
of equation
three
(3.24)
values is not
in the zero.
RM I [RM(Eo) The
integral
in equation
(3.33)
may
now
XMu,
Xku,
XMl,
upper
and
and
as the
of equation
"_k
be evaluated
RM(E°)
[exp(--aMXM "kI:'jk ,
xkl
lower
are
-- akXku
the
limits
values
of xj
terms are similarly derived. integral flux associated with
limit
of xj
for which
- x
]
(3.43)
l -- ffkXkl -- ajXjl
)
-- ajXju)]
(3.44)
of equations
(3.36)
and
(3.37)
evaluated
may
easily
show
each
term
(VM--VJ)] (t'M Vk)('kJ
may
(3.45)
be evaluated
as
dE
with
,_2)(x,E)
¢_I)(x,
equation
dE=
E)
(2.5).
E
equation 3.1.
agrees (2.6) Total
The results spectrum and 10
with has Flux
equation been
used
(3.47)
Furthermore
Iv M-
k
= °'JM [exp(-ffjx)exp(--O'MX)] o"M - aj
vk]A-----jk M
( VM _-- aVJj ) which
(3.46)
that
f0 °e
in agreement
at the
and
ffP_I)(x) = _o°° ¢_l)(x,S)dE One
the
as
[ (vk --vJ).aM Ajk M = O'j + L(VM - vk) Higher order The total
upper (3.42)
aJkak_Mff_ j
- exp(--aMXMu
corresponding
is used
- x] < E _ Rk 1
=Ek where
braces
As a result
(2.6)
as
previously
_kk
[exp(--o-jx)
vk (ref.
--*
VM.
_
[exp(-(Tjx)--exp(--akX)]
-- exp(--O-MX)]
}-_M
This
of
relation
equation
(3.48) (3.48)
and
7).
Comparisons
of equations given along
(3.28) with
and (3.44) are integrated values from corresponding
numerically over their energy-independent
entire energy solutions in
table 3.1. dependent 3.2.
The primary beam was taken solutions appear quite accurate.
Monoenergetic
Beam
as 2°Ne at 1380 MeV/amu.
Clearly,
the energy-
Results
The fluorine spectral flux seen at various depths in a water column is shown in figure 3.1. The primary beam was 2°Ne ions at 600 MeV/amu corresponding to a range of 30 cm. There is a clear structure due to the fluorine isotopes shown in the spectrum. The most energetic ions are 19F. The lSF and 17F spectral components are clearly resolved. Only the 19F is able to penetrate to the largest depth represented (35 cm). A similar, but more complicated, isotopic structure is seen in the oxygen spectra of figure 3.2. The greater number of oxygen isotopes contributing has a smoothing effect on the resultant spectrum. This effect is even more clearly seen in figure 3.3 for the nitrogen isotopes. Some of the smoothness results from the higher order term ¢(2) in the perturbation expansion. The boron flux of figure 3.4 shows very little isotopic structure. Qualitatively, similar results are obtained for an iron beam of the same range (30 cm) as shown in figures 3.5 to 3.9.
Table
3.1.
Total
Flux
From
Energy-Independent
[Values
in parentheses
Solution
are
from
and
energy-independent
Flux, Fragment
Term
18 F
¢(1)
170
q_(2) ¢(1)
16
0
15 N
13C +-2C II B
Numerically
cm -2,
Integrated
Differential
Spectrum
solution]
at water
depth
x of--
5 cm
20 cm
0.00727
(0.00717)
0.01148
0.00018
(0.00018)
0.00114
(0.00114)
0.00729
(0.00729)
0.01173
(0.01174)
0.00017
(0.00017)
0.00112
(O.QQ112___
0(1)
0.01350
(0.01349)
0.02193
_(2)
0.00029
(0.00029)
0.00191
(0.02202) (o.omgoJ_
¢(1)
0.00470 (0o0481)
0.00796
(0.00796)
0.00032
0.00220
_(2) ¢({) ¢(2) ¢(U ¢(1)
(0.0114O)
0.00511 (0.00521)
0.00894
(0.00220) (0.00887)
0.00032
(0.00033)
0.00224
(0.00224)
0.00668
(0.00682)
0.01173
(0.01178)
0.00056 (o.ooo86)
0.00398
(0.00398)
0.00417
0.00735
(0.00732)
0.00259
(0.00259)
(0.00033)
(0.00417)
o.ooo36 (o.ooo36)
11
5 _10 -3 x, cm 5 E
4--
3-i O4
'E 15
O
2-¢O
E
25
1-35 I 100
0
Figure 3.1. Fluorine column at various
flux spectrum depths.
]_ 200
produced
I
.,
L
300 400 E, MeV/amu
by
a 2°Ne
I 500
I 600
beam
at
600
I 700
MeV/amu
in a water
5 x10-3
= E "T, > _
4
x,cm 5
a
i
'E --,
2
w-t-
--
15
t
M
o
1
__ 0
Figure
3.2.
column
Oxygen at various
1O0
flux
spectrum
200
produced
300 400 E, MeV/amu
by
a 2°Ne
J
I
I
500
600
700
beam
at
600
depths.
12
FIIi
MeV/amu
in a water
5 x10 -3
4
-
E
304
'E 0
2 ¢..0.,)
o
x, cm 5
1 -
z
25 0
1O0
200
15
300
f'_
400
t
500
600
I 700
E, MeV/amu
Figure 3.3. Nitrogen column at various
flux spectrum depths.
produced
by a 2°Ne beam
at 600 MeV/amu
in a water
5 x_10 -3
E
cq 'E eD
m ¢-
o O
fD
B
x, cm 25
0
1O0
Figure 3.4. Boron flux spectrum at various depths.
200
produced
300 400 E, MeV/amu
15
5
500
600
by a 20Ne beam at 600 MeV/amu
700
in a water
column
13
8.0 _10 -3
x, cm 5
E 6.4 i
> ¢1
N
4.8
'E 0
15
ff =
3.2
m
g
1.6
m
25 /
I
_J_l
200
Figure 3.5. Manganese column at various
400
flux spectrum depths.
I,,
I
,
600 800 E, MeV/amu
produced
I
I
I
1000
1200
1400
by a 56Fe beam
at
1090
MeV/amu
in a water
8.0 _10 -3
6.4 T--
>
_, 4.8 x, cm 5
E 0
3.2 E ="1
E £ " 0
15
1.6
0
Figure 3.6. Chromium column at various
I 200
J'-'&
flux spectrum depths.
400
_
I
600 800 E, MeV/amu
produced
by a 56Fe
1 1200
1000
beam
at
1090
14
il !1
I 1400
MeV/amu
in a water
8.0 xlO -3
_= 6.4 i
_, 4.8 'E -_ 3.2 E x, cm 5 ¢.-
>
1.6 15 I 200
0
Figure 3.7. Vanadium column at various
E
flux spectrum depths.
_
25 400
I I 600 800 E, MeV/amu
produced
I 1200
I 1000
by a 56Fe beam
at
1090
I 1400
MeV/amu
in a water
1.6
.q> G)
1.2
E ¢..)
.8 t,r'0
x, cm 5
41
35
I _,,'_T
0
Figure 3.8. Fluorine column at various
200
flux spectrum depths.
400
produced
1 f I _ 600 800 E, MeV/amu
by a 56Fe
I 1200
I I I 1000
beam
at
1090
I 1400
MeV/amu
in a water
15
2.0 x--10-4
--, 1.6 E i
> _
1.2
'E x --_
.8
m
x, cm 5
¢-
0
.4 -
15
0
Figure 3.9. Carbon column at various
4. Realistic
200
400
flux spectrum depths.
600 800 E, MeV/amu
produced
by a 56Fe
1000
1200
beam
at
1090
1400
MeV/amu
in a water
Ion Beams
In the previous section, We now take the incident
we assumed that a monoenergetic ion beam flux to be
beam
was present
at the boundary.
i
Cj(0,E) = _ where Eo is the nominal beam The full solution is then found uncollided flux is found to be
RM(Em)
= RM(E
) + x.
j
2A2
(4.1)
energy and A is related to the half-width at half-maximum. as a superposition of results from the previous section. The
¢(A0/)(x, E ) = SM(Em) _M(E ) where
exp
exp(_aMx
One similarly
)
arrives
1 _-_exp[
[(Eo-
Era) 2 ]
J
5-_
(4.2)
at /
¢_l)(x'E)
=
~ °'jMVj Sj(E) IvM -
.jl
exp
-_aj
x-_l[erf(Eu-_2_°)-erffEI-E°_]\
[x - Rj(E)
_¢'_A
- _o] -
_aM 1
Ix +
Rj(E)
+ _;]
(4.3)
]]
where (4.4)
(4.5)
711o=
21,,M VM -- Yj
RM(E°)
v M _ vj [Rj(E) YM vj
+x]
16
: ltli
(4.6)
The second collision
contribution
to the ion energy
spectrum
is similar:
_kMvj _2_(x, _) : E_ _(_) O'jk...-.f_,=_ _,_M[exp((rMXM
-- exp(--_r
MX Mu
-- O'kXku
-- o'jXju)
(Eu Tc_A - Eo_)-erf\ x _1 [erf _,
l -- O'kXkl
-- 17jXjl )
]
(El_¢_- EoZl ]]
(4.7)
where (VM
,-,.-.]
El =
--,x)]
E_= Rk, [._(R_(E)+x) [
RM 1 [RM(E
>
Vk >
(v k > VM
(PM
>
Pj)
>
vj)
Vj >
Vk)
(_k> I'M> I'j)
(4.8)
(4.9)
I'M
) + x]
(I'M
> I'j
>
I'k)
and x M and x k evaluated at the upper and lower limit values of xj are obtained from equations (3.31) and (3.32). The elemental flux spectra were recalculated for 2°Ne ions at 600 MeV/amu with a 0.2-percent energy spread assumed for the primary beam. The resulting fluorine flux is shown in figure 4.1. Although the spectral results are quite similar to the monoenergetic beam case, there is a considerable smoothing of the total spectrum. Similar results are obtained for the oxygen flux as well in figure 4.2. In distinction, the nitrogen and carbon spectra show only slight isotopic structure as seen in figures 4.3 and 4.4. Qualitatively similar results are obtained for the 56Fe realistic beam as shown in figures 4.5 through 4.9.
17
5 10 -3
E
x,cm 5
4--
,v-
3-i t_
'e 5
O
2¢O
35
Fluorine energy
I 1O0
0
Figure 4.1. 0.2-percent
25
1 -
E
, 200
I
300 400 E, MeV/amu
flux spectrum produced spread in a water column
i
I
I
500
600
700
by a 2°Ne beam at various depths.
at
600
MeV/amu
with
5 x10-3
= E
4-
, >
_ a/
x, cm 5
a-
15
C
M
0
1--
25 35
0
100
_. 200
Ir 400 300 E, MeV/amu
Figure 4.2. Oxygen flux spectrum produced by a 2°Ne beam energy spread in a water column at various depths.
500
,
I
I
600
700
at 600 MeV/amu
18
lli
with
a 0.2-percent
a
5 xl0 -3
= E
4
> I1) _
3
E ff -"
2
t-
x, cm 5 35 -------P_,'f""T""--_ 0
Figure 4.3. 0.2-percent
Nitrogen energy
100
flux spread
25 200
"_ ",-J_..
15
300 400 E, MeV/amu
spectrum produced in a water column
_'_ I tL 500
by a 2°Ne beam at various depths.
I
I
600
700
at
600
MeV/amu
with
_10-3
= E
4
_
3
E ff -,
2
¢0
o
1 _
x, cm 5 25
35 0
100
200
15
I ]',,I 300 400 E, MeV/amu
Figure 4.4. Carbon flux spectrum produced by a 2°Ne beam energy spread in a water column at various depths.
.\_1 J 500
/
I 600
at 600 MeV/amu
J 700
with
a 0.2-percent
lg
a
8.0 _10 -3
E
6.4
T--
> x,cm 5
4.8 E O
3.2 {D
15
¢-
¢-
A
1.6 25
I 0
200
Figure 4.5. Manganese flux 0.2-percent energy spread
Jk,i 400
spectrum in a water
600 800 E, MeV/arnu produced column
1000
by a 56Fe beam at various depths.
I 1200
at
1090
I 1400
MeV/amu
with
a
with
a
8.0 x10"3
E 6.4 "T, ID
_ c,/
4.8
'E
x,cm 5
U
_E E £ " 0
3.2
15
1.6 25
I 200
Figure 4.6. 0.2-percent
20
Chromium energy
flux spread
,.r"'k 400
spectrum in a water
600 800 E, MeV/amu
produced column
I 000
by a 56Fe beam at various depths.
I 1200
at
1090
I 1400
MeV/amu
8.0 x_lO -3
2
6.4
>
_, 4.8 'E
O4
x_ -_ 3.2 E I'-
_
1.6
I 200
0
Figure 4.7. 0.2-percent
Vanadium energy
x, cm 5
_
flux spread
25 z'-q% 400
f,_15 I _ _ t 600 800 E, MeV/amu
spectrum produced in a water column
I 1000
by a 56Fe beam at various depths.
1 1200
I 1400
at
MeV/amu
1090
with
a
10 _10-4
-' E
8
,v-
> 6 E O
ff =
4
r'o --1
E
2 35
I 0
Figure 4.8. 0.2-percent
Fluorine energy
200
25
15
t 400
x, cm 5
_ __Y-F_ 600 800 E, MeV/amu
flux spectrum produced spread in a water column
1000
by a 56Fe beam at various depths.
at
I
I
1200
1400
1090
MeV/amu
with
a
21
8
:3
-
E T--
> 6cd {,.}
4-E
o
235
_ 0
Figure 4.9. 0.2-percent
Carbon energy
5. Approximate
25
_
i
200
400
Approximate
The
uncollided
600 800 E, MeV/amu
flux spectrum produced spread in a water column
Spectral
1000
by a 56Fe beam at various depths.
i
J
1200
1400
at
1090
MeV/amu
with
a
Solutions
Monoenergetic beam
x, cm 5
I . ,_
In the previous sections, the spectral to second-order collision terms. Such a representation of the transport solution. for the perturbation series. Clearly, the to which they are known, and the higher expressions of this section. 5.1.
15
solution
¢_0)(x, E)
were derived an adequate expressions to the order approximate
Beams
is taken
=¢j
solutions of the secondary ion flux three-term expansion is not always In this section, we derive approximate more accurate results would be used order terms would be taken to the
(0)
which is equal to the result in equation by noting that the energy dependence energy (refs. 4 through 7) resulting in
as
1
(x)_j(E-----_6[x
(5.1)
+ Rj(E)-Rj(Eo)]
(3.27). The first-order collision term is approximated of the exponent of equation (3.28) is slowly varying
• ¢_)(x) ¢_1)(x,E) _, Eju - Ejt
in
(5.2)
where
(5.3)
(5.4)
22
I
il I
Similarly,
= _ ajkakMg._(j.k,M)
(5.5)
where
Rk 1{RM(Eo)- z}
(V M
EJl= l nkl {_k nM(Eo)-X ) ( RM1{RM(Eo)- x}
(Pk
>
>
Vk >
PM
Vj)
>
/
(5.6)
Pj)
(_M > _j > "k)
and R-kl ( uM RM(E°)
- vjx }Uk
(V M
>
Vk >
V j)
[
I Eju =
R_
{ RM(E°)
- vJX }UM
(5.7) (Vk
>
tiM
>
PJ)
I
I
(.M > "5> "k) J
Rk-1 (_kM RM(Eo)-X} Higher order
terms
(n > 2) are taken as
(5.8)
where Eju and Ejl are given by equations (5.6) and (5.7). In all the expressions by equations (5.2), (5.5), and (5.8), the flux values are taken as zero unless Ejl < E < EjU The approximate monoenergetic be compared with the solutions
for ¢(.n) given T3
(5.9)
beam solutions are given in figures 5.1 through 5.4 and should found in section 3. The 170 flux at 20 cm of water is shown
in figure 5.1 as contributed by the first collision term. The trapezoidal (solid) curve is the exact solution for the first collision term derived in section 3. The rectangular (dashed) curve is the approximate first collision term of equation (5.2). Terms for other fragment spectra are similar to those shown in figure 5.1. The solution for the second collision contribution to the 170 flux at 20 cm of water is shown in figure 5.2. The nearly rectangular solution (dashed curve) is the approximation given by equation (5.5). A triangular spectral function of the same energy interval could yield improved results. The spectra of fragments which are much lighter than the primary beam are more accurately represented by the approximate solutions as seen in figures 5.3 and 5.4. This improvement results from the greater number of terms in the summation of equation (3.44). This leads us to believe that the higher order terms in the perturbation series can be adequately represented by the approximation in equation (5.8). This is especially true because higher order terms in many applications are only small corrections. 5.2.
Approximate
Realistic
Approximate solutions position of the approximate
Beams
for realistic monoenergetic
Cj(0, E) -
ion beams may be found beam solutions. The incident
1 [ V/-_-_Aexp
(E=Eo) 2A2
2]j 5jM
by using a superion beam is taken as
(5.10) 23
whereEo
is the nominal beam The first term is then as before
energy,
and
A is related
to the half-width
at half-maximum.
(Eo - Em) 2 ] ¢(M00) (x, E) = SM(Em) SM(E) exp(--O'MX)_ where RM(Em
) = RM(E ) + x. One similarly
¢_l)(x,
E)---¢_l)(x)_
[erf
(E_f_
1
arrives
°)
exp [
(5.11)
at
-erf(E_2E°)]
(Eju-Ejl)
(5.12)
-1
where Eu=R_
{ --_M[Rj(E) t'j
Et = RMI and Eju and Ejt are given by equations
@ 2)(x'E)
= Z
k
aJkakM
(5.3) and (5.4).
g(j,k,M){erf[(Eu-
+ x]}
(5.13)
+ z]}
(5.14)
Additional
Eo)/V"2A-
2(E u - Ej,)
computation
erf(El-
Z Jl ,..,,in-
ajj,,_, l
.....
trjl,M
g(J, Jn-1
.....
jl'M)
erf[(Eu-E°)/v_A]-erf[(E1-E°)/v/2A]
2(Ej,, - E_t)
(5.15)
Eo)/V_A]}
where Eju and Ejt are given by equations (5.6) and (5.7), and E t and equations (4.8) and (4.9). The remaining higher order terms are taken as
¢_ n)(x'E):
yields
Eu
are
given
in
(5.16)
where Ejl and Eju are given by equations (5.2) and (5.3) and Eu and E l are given by equations (5.13) and (5.14). These approximate equations for realistic ion beams are given in figures 5.5 to 5.8 and should be compared with the more exact formulae given in section 4. The primary ion beam is taken as 20Ne at 600 MeV/amu with a 0.2-percent energy spread. The 170 flux first collision term is shown in figure 5.5 for the two formalisms. The effect of the beam energy spread is seen as a rounding of the spectrum at the edges compared with the monoenergetic case in figure 5.1. The second collision term is shown in figure 5.6. The approximate second collision term improves for the lighter fragments as seen in figures 5.7 and 5.8. Higher order collision terms are expected to be more accurate because of the large number of combinations of contributing ion terms.
24
--
I I !i
x 10-4 First collision term Approximate first collision term
.... 1.5 E i T" I
>
.5 T--
0 280
Figure 5.1. term.
I 300
I 320
170 flux spectral
term
I I 340 360 E, MeV/amu
according
i 380
to first
1 400
collision
,I 420
term
and
approximate
first collision
3 xl0 -5
/ E
_
_
Second collision term
....
Approximate second collision term
2
i
>
......
-
....
{D I
,
E O
,,
_
i ,
0 280
Figure 5.2. collision
I i I i !
300
170 flux spectral term.
,,
320
term
340 360 E, MeV/amu
according
to second
380
collision
.400
term
I 420
and
approximate
second
25
x 10-5 Second collision term
ate second collision term :3
E
T. 3 -
I
C_I
'E o x"
2
:3
o ,-(.O
1
-
! .....
0 280
Figure 5.3. collision
300
160 flux term.
6-
!
spectral
I
t
320
340 E, MeV/amu
360
term
according
'1
x_
380
to second
1 400
collision
term
and
approximate
second
x 10 -5
-- Approximate second collision term
:3
--
E /
Second collision term
.......
I
i
cq
I
'E :3
T-
0 280
Figure 5.4. collision
'_ 300
14N flux term.
I
I
320
340
spectral
term
I
I
360 380 E, MeV/amu
according
to second
I
"
400
I
I
420
440
collision
term
26
il lli
and
approximate
second
-....
2.O xl0 -4
1.5
First collision term Approximate first collision term
--
E i
1.0E
.5
--
T--
0 280
_J
l
300
Figure 5.5. 170 flux spectral term and approximate first collision
320
for energy term.
I
I
I
340 360 E, MeV/amu spread
li
380
2°Ne beam
J
400
according
420
to first
collision
term
3 x10-5 Second collision term
E
2
T-
/
_--
/
\
t .....
-- Approximate second collision term
---'l
eD
¢,/
'E ¢)
--n
o
0 280
300
320
340 360 E, MeV/amu
Figure 5.6. 170 flux spectral term for energy spread term and approximate second collision term.
380
2°Ne
400
beam
I 420
according
to second
collision
27
xl 0-5 Second collision term _
/_
E t --
2
second
collision term
_
I I
CD
I I
/
m
/
O ¢.O
Approximate i I i
j
'T
tJ
__ ....
# I !
1 -#
I I
t
•
0 28o
300
320
340 E, MeV/amu
360
Figure 5.7. 160 flux spectral term for energy spread term and approximate second collision term.
1 400
380
2°Ne
beam
according
to second
collision
_10 -5 _____
E
4
Second collision term
-- ipproximate
second
collision term
i
> (D ;E
# .........
I I I ! i t i I i f I ! #
'E x"
2
z
! 1 ! I ! 1 i ! i I i I !
/ 0 28O
300
, 320
340
360 380 E, MeWamu
Figure 5.8. laN flux spectral term for energy spread term and approximate second collision term.
400
2°Ne
420
beam
t 440
according
28
1|1i
to second
collision
6. Recommended
Methods
An energy-independent theory has been used to show that the perturbation expansion up to the double collision term is adequate for all fragments whose mass is near that of the projectile. This is why the three-term expansion was able to explain the Bragg curve data for 2°Ne beams in water with reasonable accuracy (ref. 7). As a starting point for the calculation of the transition of heavy ion beams in materials, the use of the three-term expansion of sections 3 and 4 can be further corrected by use of the approximate higher order terms given in section 5. As an example of such a procedure, we give results for ZONe beams at 600 MeV/amu in water. The results are shown in figures 6.1 to 6.6 as successive partial sums of the perturbation series. The solid line is the first collision term. The dashed curve includes the double collision terms. The long-dash-short-dash curve includes the triple collision term and can hardly be distinguished from the long-dash-double-short-dash curve which includes the quadruple collision terms. The results for penetration to 20 cm of water are shown in figures 6.1 to 6.6. The monoenergetic beam results for 170, 160, and 12C are given in figures 6.1 to 6.3, respectively. The double collision term is seen to be always an important contribution. The triple collision term shows some importance for 12C, while higher order terms are negligible. Similar results are shown in figures 6.4 to 6.6 for an energy spread of 0.2 percent. NASA Langley Research Center Hampton, VA 23665-5225 May 11, 1989
29
2.5 x10"4 -2.0
-----
--
Theory First order Second order Third order Fourth order
E i
"7 :>
1.5--
_
1.0-
0 .5
/
0 280
_-,.-4----'_, . I 300 320
I 1 340 360 E, MeV/amu
1 380
Figure 6.1. Sequence of approximations of 170 flux spectrum second-order, third-order, and fourth-order theories.
_1 400
after
I 420
20 cm of water
for first-order,
5 x_lO-4 -4-
----....
Theory First order Second order Third order Fourth order
E i
v >
3 -
O
O
2--
x" "1 m
o ¢O T-
1-
0 280
300
320
340 E, MeV/amu
Figure 6.2. Sequence of approximations of 160 second-order, third-order, and fourth-order
360
flux spectrum theories.
380
after
400
20 cm of water
3O
_]
|
Ii
for first-order,
ORIGINAL
PAGE
IS
OF POOR
QUALITY
2.5 xlOJ
m
Xh_ Firm
....
_r
....
FaldN_r
2.0
E ? 1.5O
_J 1.0-
O
c_ 04 ,f-
.5
--
0 275
300
I
I
325
350
,
I
[
]
375 400 E, MeV/amu
_
425
Figure 6.3. Sequence of approximations of 12C flux spectrum second-order, third-order, and fourth-order theories.
after
J 450
475
20 cm of water
for first-order,
2.5 xlO -4
.... ----....
2.0
Theory First order Second order Third order Fourth order
E 1.5
# E
1.0
O t',T--
.5
--
0 28O
_ _L.._
I
300
320
I
I
340 " 360 E, MeV/amu
Figure 6.4. Sequence of approximations for energy spread 20 cm of water for first-order, second-order, third-order,
I
-- I
380
400
I 420
solution of 170 flux spectrum and fourth-order theories.
after
31
5 x10"4
4
-.... ----....
-
Theory First order Second order Third order Fourth order
E >
¢_
3-
2-
O ¢.D
1 -
280
300
t
I
I
_
J
320
340 E, MeV/amu
360
380
400
Figure 6.5. Sequence of approximations for energy spread 20 cm of water for first-order, second-order, third-order,
solution of 160 and fourth-order
flux spectrum theories.
after
2.5 xlO 4
----....
2.0
Theory First order Second order Third order Fourth order
E "7 >
1.5
Q)
E O
1.0
o C_ ,¢-
.5
0 275
-J"J 3OO
I 325
I 350
I [ 375 400 E, MeV/amu
Figure 6.6. Sequence of approximations for energy spread 20 cm of water for first-order, second-order, third-order,
32
I 421
•--.J 450
I 475
solution of 12C flux spectrum and fourth-order theories.
after
References 1. Wilson,
John
1986,
pp.
W.;
and
Badavi,
2. Wilson, J. W.; Townsend, Atmosphere. Radiat. Res., 3. Wilson,
John
W,; and
Radiat.
Res.,
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J.
of Galactic
L. W.; and Badavi, vol. 109, no. 2, Feb.
Townsend,
114, 1988,
W.:
John
6. Wilson, TP-2178,
Lawrence
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Depth-Dose
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John 1983.
7. Wilson,
John
Jerry:
W.;
Barry
Transport
9. Townsend, Nucl. Soc., Shavers, 670A
W.:
A Benchmark
for Heavy
Ion
Transport.
Radiat.
P_'opagation
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Townsend,
Theory Ion
D.; Townsend,
Virginia
Lawrence
Energy
R.:
Heavy-Ion Neon
Health
W.; and
and Spatially
the
Wilson,
Cross
Code
Interacting
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Res.,
Through
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the
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3,
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H. H.:
Transport.
Straight
Dependent
Nuclear
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H. B.; Schimmerling,
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L. W.; and Wilson, J. W.: vol. 56, 1988, pp. 277-279. Accelerated
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L. W.; Bidasaria,
Relations
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Mark
MeV
of the Heavy
Ziegler, J. F.; Biersack, J. P.; and Littmark, The Stopping and Ranges of Ions in Matter,
16.
Ion
F. F.: Galactic HZE 1987, pp. 173-183.
11.
14.
Heavy
206.
Relations
Analysis W.:
2°Ne Depth-Dose
8. Ganapol,
10.
Methods
136-138.
5. Wilson,
Ion
F. F.:
231-237.
Pergamon
Ions
in All
Press
Inc.,
Elements. c.1980.
33
Report Natio_a? Spa_e
Aeror_autics A_ministr
1. Report
No.
[ 2. Government
Page
and
Accession
No.
3. Recipient's
Subtitle
5. Report
A Hierarchy of Transport Heavy (HZE) Ions
Approximations
for High Energy
7. Author(s) John W. Wilson, Stanley L. Lamkin, Hamidullah Barry D. Ganapol, and Lawrence W. Townsend 9. Performing
Organization
Name
and
Agency
Name
and
Organization
Code
8. Performing
Organization
Report
Farhat,
Supplementary
10. Work
Address
Unit
No.
199-22-76-01 11. Contract
Address
or Grant
of Report
Technical 14. Sponsoring
No.
and
Period
Covered
Memorandum Agency
Code
Notes
John W. Wilson and Lawrence W. Townsend: Langley Research Stanley L. Lamkin: PRC Kentron, Inc., Aerospace Technologies Hamidullah Farhat: Hampton University, Hampton, Virginia. Barry D. Ganapoh University of Arizona, Tucson, Arizona. 16.
No.
L-16572
National Aeronautics and Space Administration Washington, DC 20546-0001 15.
Date
6. Performing
13. Type Sponsoring
No.
July 1989
NASA Langley Research Center Hampton, VA 23665-5225
12.
Catalog
I
NASA TM-4118 4. Title
Documentation
a_J
aElOn
Center, Hampton, Division, Hampton,
Virginia. Virginia.
Abstract
The transport of high energy heavy (HZE) ions through bulk materials is studied with energy dependence of the nuclear cross sections being neglected. A three-term perturbation expansion appears to be adequate for most practical applications for which penetration depths are less than 30 g/cm 2 of material. The differential energy flux is found for monoenergetic beams and for realistic ion beam spectral distributions. An approximate formalism is given to estimate higher order terms.
17. Key
Words
(Suggested
18. Distribution
by Authors(s))
Nuclear reactions Heavy ions Transport theory
Statement
Unclassified-Unlimited
Subject 19. Security
Classif.
(of this
FORM
1626
93
report)
20. Unclassified Security Classif.(of this page)
Unclassified _TASA
Category
121"N°" 37 °f Pages 122"Price A03
OCT 86 For sale
NASA-LanglPy,
by the
National
Technical
Information
Service,
Springfield,
Virginia
III
22161-2171
1989