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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963

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'ORNLY 299
DIE
MASTER
SEP 2 1 1984
NUMERICAL SOLUTIONS OF THE ONE-DIMENSIONAL
NUCLEON-MESON CASCADE EQUATIONS*
R. G. Alsmiller, Jr.
(Oak Ridge National Laboratory
Oak Ridge, Tennessee
J. E. Mirphy**;
J. Barish
Central Data Processing Facility
(Oak Ridge Gaseous Diffusion Plant
Oak Ridge, Tennessee 112 dana

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*Research sponsored partially by the U. S. Atomic Energy Commission
under contract with the Union Carbide Corporation and partially by the
National Aeronautics and Space Administration under NASA Order R-104.
**Present address: Arthur D. Little Company, Boston, Massachusetts.
TIF.P.
NUMERICAL SOLUTIONS OF THE ONE-DIMENSIONAL
NUCLEON-MESON CASCADE EQUATIONS*
R. G. Alsmiller, Jr.
Oak Ridge National Laboratory
Oak Ridge, Tennessee
J. E. Murphy**
J. Barish
Central Data Processing Facility
Oak Riäge Gaseous Diffusion Plant
Oak Ridge, Tennessee
Abstract
In shielding calculations for high-energy accelerators it is nec-
essary to solve the nucleon-meson cascade equations numerically for
very large distances. For the case of a 10-GeV proton beam and a set
of quite special physical assumptions, an analytic solution has been
obtained and compared with the numerical solution. The two solutions
are shown to be in excellent agreement for thicknesses as large as
30 collision mean free paths (w2800 g/cm).
-*Research sponsored partially by the U. S. Atomic Energy Commission under
contract with the Union Carbide Corporation and partially by the National
Aeronautics and Space Administration under NASA Order R-104.
**Present address: Arthur D. Little Company, Boston, Massachusetts.
'
-3-
I. Introduction
In a series of reports numerical solutions to the equations describing
a one-dimensional nucleon-meson cascade have been given for a variety of cases
of interest in the shielding of manned space vehicles and high-energy acceler-
ators.114 In the case of accelerator shielding where very thick shields are
involved, the numerical calculations are quite extensive and the truncation
error could be excessive.
For a very special case an analytic (quadrature) solution to the cascade
equations has recently been obtained.5 In this paper the numerical solution :
and the analytic solution are compared for the case of a 10-GeV proton beam
and are shown to be in excellent agreement after a shield thickness of 30 col-
lision mean free paths.*
In Section II the cascade equations are given. In Section III the assump-
given. In Section IV the comparison between the analytic and numerical solu-
tions is shown.
II. Cascade Equations
In writing the cascade equations we shall neglect neutral pions since
they decay very rapidly into two photons and photons are not included. Further-
more, no distinction will be made between positively and negatively charged
pions, and charged pion decay will be neglected.**
Under these circumstances the one-dimensional cascade equations for the
particle fluxes may be written®: ?
,
j = N, P,05 ,
(2.1)
.
FICE
d
STEJ = r o
(2.2)
0031831) + e(€) 0,318, 1) - de ($ (3) Og(8,v]
- Jos 12") ©_ca"(165,3 +6"]=",
Fik(E', E) Qiel
sklE',r) + z3(E',
3, k = N,P, a
,
(2.3)
where
....
N,P,1 = neutron, proton, and charged pion, respectively,
.....
en
....
..
.
--*...*..
0,,(E,r) = primary flux per unit energy range of particles of type j,
OS (E,r) = secondary flux per unit energy range of particles of type 3,
.*
*-.-
E = kinetic energy,
?-
:
!
r = dimensionless distance variable defined by the relation r = SR,
üdwesty
o = density of medium, in g/cms,
R = distance, in cm,
S
nemen
i
do = an arbitrary constant with dimensions g/cm2 which determines the
units in which r is measured,
Q; (E) - SAMO O,(E),
No - Avogadro's number,
0,(E) - nonelastic cross section for particles of type j in the medium
being considered,
A = atomic weight of medium being considered,
S; (E) =
0 €, (E),
€ (E) = stopping cross section for particles of type 3,
F(E', E) LE = the number or secondary particles of type j in the energy interval
E to E + de produced by the nonelastic collision of a particle of
type k with energy E'.
.
III. Physical Assumptions and Analytic Solution
To reduce the equations to soluble form, we introduce the assumptions
Q:(E) = Q = constant,
j = 1,P, 17 ,
(3.1)
S,(E) = S = constant, j = P,
,
(3.2)
Sy(E) = 0,
(3.3)
F13(E',E) = Chi Yz els
eV(E'-E) (E' - E), 1, j = 1,P,
..
(3.4)
Olgy ?, v = constant ,
(E' - E) = 1
E' ? E
= 0
E' < E
.
These assumptions are, of course, introduced for their simplicity and represent
only very approximately a real physical system. In particular, Eq. (3.4) is not
very realistic. With appropriate choice of v the E dependence may be made
physically reasonable, but the resulting particle multiplicity varies much too
rapidly with E'.
With the assumptions of Eqs. (3.1) to (3.4) a quadrature solution to
Eqs. (2.1) to (2.3) can be obtained. The details of obtaining the solution are
given in Ref. 5, so only the solution itself will be given here. Furthermore,
since only the case of a monoenergetic initial proton flux is of interest here,
only the solution «ppropriate to this case will be given.
Using the initial fluxes
R(E) = Po 8(Eo - E) ,
03(E) = 0,
j = N,5 ,
(3.5)
where Po = constant and Eo = energy of incident protons, the solution may be writ-
ten
-
-
-
Qap(E,r) = Po 8(E. - E - Sr) ell
,
(3.6)
Øsp(E,F) = Po app et K(E. - E,r),
(3.7)
Srt (E, r) = P. On peut K(E. - E, r) ,
(3.8)
sy( 8, z) • Po
? •
ਬ(zo • E,r) KE • ) ,
(39)
where
v(En-E-Sr)
•
6(E) • # • sr)
ਲ, ਵਰ · ਗ਼ਵਰ 1, ਪਾਲ - - - -
| • 1928 de, - ਝ - 6:)
* ਵਾਂ , - : - az #ar) { w[V - 1 - it is a
[: : : : : : : * 3 ਆਜ - - : • ਬਟਾਂ ਦਾ ])} ,
• ॥
. Sr +
on
E
- Sr of Sr' - S
r By(s • : • Sr + Sr' - Sr"
r .
(zo •
• Sr + Sr' • Sr") (r
.
(3.10)
ਗੱਲ - ਜੇ - Jਵਾਂ , -ਸ: -1 -
· 1. [e Vya - - r) (r - ]
y(so-E-Sr')
(z
• # - Sr')
,
(31)
ਖ਼ = % ੧ ,.
.
* (
? + , ( 4 ,
(3.12)
and Io and Iq are the usual hyperbolic Bessel functions..
-8.
IV. Comparison of Numerical, and Analytic Solutions
In doing the numerical computations the constants appearing in the equa-
tions were chosen to be
Q; = 1,
j = N, P,R
,
S, - 187.6 Mell,
J = P,n
,
Sn = 0,
v = 7 x 10-4
ag = 10^2
j = N, P,R
,
1y = 1/v.
j = N, P, T
,
10 = 93.8 z.'coma o
and the constants in the initial flux were chosen to be
Po = 1 pro./comsec,
.
Eo = 10 GeV
.
Before giving the comparison, it is perhaps worth while to make one
point. In obtaining the numerical solutions all calculations are done in terms
of a lethargy variable, u, defined by
u = 108 [!
u = log
---
and in terms of lethargy
.... w...
stw =
1.e., the stopping power 18 not constant. ***
Thus in doing the calculations a
variation of the stopping power in lethargy was taken into account.
Prom Eqs. (2.1) and (3.6) It follows that the calculation of the primary
flux in the present Instance is quite trivial. The two calculations give for
all practical purposes the seme answer, and therefore a comparison of the primary
flux is not given.
In Fig. 1 the secondary neutron flux as a function of energy for various
r values 18 shown, and in Fig. 2 the secondary proton flux as a function of energy
is shown. The solid curves represent the numerical solution, while the plotted
points represent the analytic solution. Because of the manner in which the con-
stants are chosen, the comparison of the pion fluxes 18 exactly the same as the
comparison of the proton fluxes and is therefore not shown.
The two solutions are in excellent agreement at all energies and all r
values considered. At the very high energies' (all. curves go to zero at 10 GeV)
the numerical solution for the proton flux tends to be slightly higher than the
analytic solution, but, since the spectrum is decreasing so rapidly at these
energies, the error 18 of no practical importance.
Acknowledgement
It is a pleasure to thank Dr. F. S. Alsmiller for many helpful discus-
sions concerning both the numerical calculations and the analytic solution.
10.
Footnotes
*The IBM code which was used in doing the numerical calculations reported here
18 an improved version of the code used before;24 however, it does not give
appreciably different results from those obtained previously.
**This decay 18 neglected here because the analytic solution can be obtained
only under this assumption. In general, our numerical solutions include this
decay and the resulting muon component.
***The cascade equations written in terms of lethargy are given in Ref. 2.
-11-
References
1. R. G. Alsmiller, Jr., and J. E. Murphy, Space Vehicle Shielding Studies:
Calculation of the Attenuation of a Model Solar Flare and Monoenergetic
Proton Beams by Aluminum Shields, ORNL-3317 (1963); Part II, The Attenuation
or Solar Flares by Aluminum Shielde, ORNL-3520 (1964); Part III, The Attenua-
tion of a Paz ticular Solar Flare by an Aluminum Shield, ORNL-3549 (1964).
2. R. G. Alsmiller, Jr., F. S. Alsmiller, and J. E. Murphy, Nucleon-Meson Cas-
cade Calculations: Transverse Shielding for a 45-GeV Electron Accelerator,
Part I, ORNL-3289 (1962); Part II, ORNL-3365 (1962); Part III, ORNL-3412 (1963).
3. R. G. Alsmiller, Jr., and J. E. Murphy, Nucleon-MeBon Cascade Calculations:
The Star Density Produced by a 24-GeV Proton Beam in Heavy Concrete, ORNI-
3367 (1963).
4. R. G. Alsmiller, Jr., and J. E. Murphy, Nucleon-Meron Cascade Calculations:
Shielding Against an 800-MeV Proton Beam, ORNL-3406 (1963).
5. R. G. Alsmiller, Jr., A Solution to the Nucleon-Meson Cascade Equations Under
Very Special Conditions, ORNL-3570 (1964).
6. H. Messel, Progress in Cosmic Ray Physics, Vol. II, Interscience Publishers,
Inc., New York (1954), p. 197.
7. B. Rossi, High-Energy Particles, Prentice-Hall, Inc., New Jersey (1956),
p. 232.
.12.
figure Captions
Fig. 1
Secondary Neutron Flux vs. Energy (Ep = 10 GeV). - Numerical
solution; * analytic solution; r 18 measured in collision lengths
( = 93.8 g/cm2).
Fig. 2
Secondary Proton Flux vs. Energy (E. - 10 GeV). — Numerical
solution; X analytic solution; r 18 measured in collision lengths
( = 93.8 g/cmt).
Ns (E, 7) {(neutrons/MeV-cmsec)/(proton/cm?-sec)]
10
102
-
-
E (MOV)
103
AN
NO
ORNL-OWO 64-6243
UNCLASSIFIED
104
NG (E,) ((neutrons/MeV.cm-sec)/(proton/cmsec)]
10-19 L
10948
1017
10916
10-15
109
102
E (MeV)
00000
-
ORNL-DWG 64.6244
UNCLASSIFIED


INO
000
Ps (E, ) [(protons/Mev.cm?-sec)/(proton/cm²-sec)]
to 100
10°12
.
.
-.
-.
:.!
10²
CON
E (MeV)
103
UL
ORNL- OWO 64-6148
UNCLASSIFIED
104
Ps (E, ) [(protons/Mev.cm²-sec)/(proton cm²-sec)]
10*20
109
E (MeV)
.
C
ORNL-OWO 64.6147
UNCLASSIFIED


HIIN
111
.
END
T-
DATE FILMED
18 / 31 /65






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