(HZE) Ions - NASA Technical Reports Server (NTRS)

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


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


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'-'&


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


_

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


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


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


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


20

Chromium energy

flux spread

,.r"'k 400

spectrum in a water

600 800 E, MeV/amu

produced column

I 000


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


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


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


Fluorine energy

200

25

15

t 400

x, cm 5

_ __Y-F_ 600 800 E, MeV/amu


1000


at

I

I

1200

1400

1090

MeV/amu

with

a

21

8

:3

-

E T--

> 6cd {,.}

4-E

o

235

_ 0


Carbon energy

5. Approximate

25

_

i

200

400

Approximate

The

uncollided

600 800 E, MeV/amu


Spectral

1000


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


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 /


.......

I

i

cq

I

'E :3

T-

0 280


'_ 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


-- 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-

----....


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


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

-.... ----....

-


E >

¢_

3-

2-

O ¢.D

1 -

280

300

t

I

I

_

J

320

340 E, MeV/amu

360

380

400


solution of 160 and fourth-order

flux spectrum theories.

after

2.5 xlO 4

----....

2.0


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


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.,

vol.

4. Wilson, pp.

J.

of Galactic

L. W.; and Badavi, vol. 109, no. 2, Feb.

Townsend,

114, 1988,

W.:

John

6. Wilson, TP-2178,

Lawrence

pp. 201

Depth-Dose

W.:

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

for Galactic

Beams.

Townsend,

Theory Ion

D.; Townsend,

Virginia

Lawrence

Energy

R.:

Heavy-Ion Neon

Health

W.; and

and Spatially

the

Wilson,

Cross

Code

Interacting

Phys.,

Res.,

Through

vol.

the

Cosmic-Ray

Transport

J.

28,

Sci.,

108,

Earth's Codes.

vol.

no.

3,

Water.

13.

The Stopping and Ranges of Ions in Matter, J. F. Ziegler, Andersen, H. H.; and Ziegler, J. F.: Hydrogen--Stopping of The Stopping Ziegler,

J. F.:

and Ranges 15. Ziegler, Stopping Littmark, Volume

Bibliography

and Ranges Helium

of Ions

J. F.: and

and

U.; and 6 of The

o.f Ions

Stopping in Matter,

Handbook Ranges

1977,

Ziegler,

Powers

in Matter, J. F.:

and

pp.

for the Galactic

1989.

Transport.

Comparison

Trans.

With

Univ.

Ranges

Cross-Sections

and

Stopping

organizer, Press

for Energetic

Ions

ed., Pergamon

of Range in Matter,

Distributions J. F. Ziegler,

Power

Inc.,

Data.

Press

Inc.,

Volume

Inc.,

Volume

1 of

Volume

2 of 3

c.1977.

4 of The

Stopping

Volume

5 of The

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for Energetic ed.,

a

c.1977.

in All Elements.

Press

for

1988.

Press Inc., c.1977. in All Elements. Volume

Matter.

Pergamon

American

Experiment

of Florida,

Pergamon

in All Elemental

J. F. Ziegler, o] Ions

Range

and Howard,

1101-1111.

Solutions TP-2878,

organizer, Pergamon Powers and Ranges

J. F. Ziegler,

organizer,

Handbook

and Ranges

NASA

Thesis,

1977. NASA

Mervyn;

1984,

Benchmark

and M.E.

of Experimental

J. F. Ziegler,

W.:

D-8381,

U.: The Stopping and Range of Ions in Solids. J. F. Ziegler, ed., Pergamon Press Inc., c.1985.

in Matter,

of Stopping

o.f Ions

Stopping

Index

Wong,

for Hadronic

Calculations

Andersen,

TN

Approximation.

Walter;

Problems.

Sections

With

NASA Ahead

vol. 46, no. 5, May John

12.

H. H.:

Transport.

Straight

Dependent

Nuclear

Transport Beam

Ion

in

H. B.; Schimmerling,

in Water.

L. W.; and Wilson, J. W.: vol. 56, 1988, pp. 277-279. Accelerated

of High-Energy

Transport

L. W.; Bidasaria,

Relations

Equations:

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