ORIG. RUSSIAN: METHODS OF SOLUTION OF TRANSPORT EQUATION IN INHOMOGENEOUS AND FINITE MEDIA

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP88-00904R000100110011-7
• Release Decision: RIPPUB
• Original Classification: U
• Document Page Count: 16
• Document Creation Date: December 22, 2016
• Document Release Date: August 26, 2009
• Sequence Number: 11
• Publication Date: May 1, 1964
• Content Type: STUDY

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[?9] is useful. Further, we use the notation of this report.
Zet ~ (Q) be a square summable with weight P(Q)(PC4,) ~' 0) in
D function (Q E D).
Table 1
PAR
go. $ampl. ins
C ~. ___
~6 ~
1
,_ _
0.0678
.
0.06 56
2
0.0689
0.0660
3
0.0624
0.0634
4
0.0843
0.0721
5
0.0596
0.0637
6
0.0688
0.0628
?
0.0842
0.0689
8
0.0746
0.0660
9
0.0896
0.0761
10
0.056?
0.0615
Average
0.0709
0.0666
Baact value
0.0665
Consider a generalized interpolating polynomial
p(~)=~~~P~(~)?cs~s(Q)ficN~-,~Q) ~ constants ~?~ (ig0~1~...,n)
are defined from the conditions
Rhere Qd are certain fined points o~ a region D. The tollo-
wing theorem is valid..
Theorem 1. If the points Q are randomly sampled in
D with the tensity of a probability~/? (G?a~Q,~ ~ ... ~ Q-, ),
Wh -(n+i~ ~ [ref II ~o ~Qm) P(4~)~... ~~h (Q,~) P~ (~ ~~o ,~ ~ ~
then
IInder the same assumptions it is valid.
Theorem 2. A standard deviation of the random value C~
is equal to
n 2
"p ~~0
a
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The results of the ealculat ion of a numerical gamma
spectrum in the air ~y the local calculation method are pre-
sented in the Table 2.
The similar values obtained by a linear algebraic in-
terpolation by random points are prese*~t~d in the Table 3.
In a given case the weight function is not introduced.
The four groups of data are presented in these tables.
Each group is obtained as a result of 1 X000 tests.
A point isotropic monoenergetic source with Eo~1.25Bdev
is used as a gamma-ray source. The source-detector distance
is 30 meters. 0-1.25Mev energy interval is divided into
eight uniform g~youp s. I f the quantity pre sent ed in the t ab-
les 2 and 3 multiplie2 by 10-9~ then We obtain a number of
gammar-rays /second/cm ~ which is belonged to tie given energy
group.
No. of sampli~;
No, o~ group
Table
1
1.006
0.864
C.678
2
1.463
1.359
0.910
3
0.335
0.437
0.753
4
0.301
0.127
0.117
5
0,461
0.415
0.289
6
0?398
0.095
0.281
7
0.192
0.050
0.694
8
0.000
0.417
0.087
Integrated flux
_.
3.986 ,
x.763
3.809
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1
2
3
4
1.580
0.925
0.73?
0.808
1.039
1.162
1.123
0.975
0.450
0.296
0.342
0.496
0.82E
0.303
0.104
0.228
0.123
0.105
0.214
0.149
G.138
0.594
0.376
0.129
0.053
0.145
0.110
0.112
0.073
0.208
0.164
0.022
4.832
3.322
4.107
2.91?
0.82?
1.322
0.173
0.292
0.576
0.356
0.000
0.460
41006
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Calculation of the functionals from the flux
1. Monte Carlo method is often used for calculation of
the functionals f from the flux ~ (Z X12 ~ E) in a space-
energy region in the inhomogeneous problems of a neutron
transport theory
~' f d-n df clEo f cl zo S(`~n,-~-a, Eo~SdnJ dEfdi f (i,,~,E~c~(ioF.~; z,~-,E~
aEn aVo aE av {20)
where ~ (Z ~.~ ~ E ; Ze~.~1.~~ E,,) is a Green funotion of the cor-
responding transport equati on. As a rule the regions of de-
termination of the source S {Ze ~_fL ~ Ee) and the function
f- (Z ~..(1 ~ E) are greatly differed ~ therefore ~ the increase
of a statistic efficiency is achieved When a sampling of the
original coordinates of history is performed in accordance
with a ad joint function in relation to functional ~10] .
Accurate finding of a ad joint function is nor less eompleg
problem than the original one ~ therefore for evaluation
of the functional ~ it is naturally to try to construct
such adjoint function. Ti' this function (~ (Zo~.lLe~ Eo) is
constructed the functional ~ may be written in a following
manner ~ 1 ] ~ ~ ~ ~ ~,- ~,.
?~0~11-d J^~Eoso~~,o S (^!3-c,~c, Eo~ R { Zo, `R'o , Eo) x
(21)
.. ~.
w erer -. -. S(~e?Ro Eo) Q CZo -moo, Eon _
.~ ~ `~o,n-o, EeJ ' J'c,1.JLnSo~Eosd'Go S (~c,~o, Eo~ (~i (~o,~'o,Eo~ J
R ~Zo,-~o,~01 ~ .sol.n..fdE f oit S? Q
*w 'Le, . Fe 1
and sampling of origina coordinates of the history may be
performed in accordance with density S and initial Reight R
iAhen determining the parameters of the resolved neutron
levels according to the results of the radiation oapture
measurements the necessity is apps~ared to calculate the f~n-
etionals of a type {20) Whioh represent in this case the
average absorption rate in a planar sample o3 thiokness H
of the neutrons from a plane source With a given spectz-nm
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S(Eo) in the energy interval a Eo entirely comprising the
resonance considered (11 ~12~13). In this problem Monte Car-
lo method 1:,s used for evaluation of the average absorption
rate of neutrons suffering scattering Ps which has the
following form : ,~
(~=,f c.~~,, s (E~),.f ~~1 cIEJd~~?(E~56 (~,l~,E,Eo~
s eEd -~ aE ~ d (22)
where C~s (~ ~ ~i ~ E , Eo) is a neutron flog after a single scat-
tering. The application o~ a ad~oint function for estimati-
on of a functional (22) is of importance as a region of
determination of the source Q So always co~piderably exoeeds
the region where a ad~oint functioa W(Eo) J~folE fdzF~(E~~(z,N,E,Eo~
-~ ~E o
is differed from zero.
For a construction of the approximate function C~ (E,~
take a following model of the neutron transport. Zet us be-
lieve that the neutrons diffuse-through a sample of an effe -
etive thickness N~ without a collision but at the boundary
some fraction of neutrons is reflected with an energy change
by the magnitude of an average logarithmic loss ~ ~
Tien an approximate ad~oint ~unetion is written down
in the~~ollowin.g manner:
where Ktv wHr ro ~ Gs (E~) -H,~ (EK)
p - C ~~ ~t r
The normalized ad~oint function is always used in an
importance sampling of original coordinates ~ therefore the
deviations of parameter values H ~ and P from H and Z p
respectively do not influence greatly on the efficiency of
approximate function applioation.
The main feature of this function - a maximum displace-
ment relative to a resonance energy - is well interpreted
by introduotion of an average logarithmic loss ,~ intQ the
expression (23), In the case of realisation of Monte Carlo
method for estimation of the funct ionals of a type (22) it
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1-C,~~ respectively.
The appl i cat ion of the method of splitting i s highly
useful in the case of Conte Carlo calculation of the influ-
ence function for monoenergetic isotropic source located
at the block surface in the infinite cylindrical lattice.
This funet ion being analogous to the Plachek' s function in
a homogeneous medium is interest in the resonance absorption
problems in heterogeneous media.
In this problem the following quantity was considered
as a functional ~ r.
representing the average density of collisions in a block
in the energy interval E < < E ~ E ~ + ~
As a rule the determination of boundaries of a multidi-
mensional region Cry and sampling of random point from this
region is a very difficult tasks therefore it is desirable
to arrange the splitting by one of the variables.
In the case considered a region ensuring the contribu--?
tion to the functional ~ is defined by the following condi-
tion: II R s~
~ ~)
where f ,,{, is a azimuthal angle of the neutron velocity direc-
tion after scattering; Rsn is a block radius. The splitting
is performed by one variables ~l-1 ~ a cosine of scattering
angle in a system of inertia. ih8.i~ ~ one can obtain the fol-
lowing rati~s for weights of the branches.
G+~ a I~o,~_~oe)~, G~ ~ ~-Gi ,
where ;Lt ~ ~= ~,~ (M-' ~.~ xLL Si.~z O+Mzn~~.. xL~ SiN2~
o~~i
M (stir D x'; ~, + 2L ~.
ah31e 7,~~,=.fJ,~ -+ 1~+~~ ~ ?~ i ,x,,L =.fl COS ~ t 7-p~ SiH f ,
~~= Si.t1o4 ~i1~,f,,+CO? v( C0~?1N C0~ Qn j ~,t Sc,H'oS cosfx - ~c7Si.~ ? frkfM co.s Q,,
M- is a nuclear mass.
Calculation of a heterogeneous Plachek's function showed a
considerable improvement of a statistic efficiency of Edon--
te Carlo method at the expence of application of the split-
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tang techniques particular in a case oT very thin blocks.
REFERENCES.
1. V.Fano, Z. Spencers M. Berger Gamma-Ray Transport ~ M.
Gosatomizdat ~ 1963.
2. G.T.Marchuk. Methods oT Calculation of Nuclear Reactors,
M. , Gosatomizdat ~ 1961.
3. B.Karlson, Numerical Solution oT Problem on Neutron
ginetic Theory. Coll o~ articles "Nuclear Reactor Theory"
edited by G.BirkhoTT ~ E.Vigner, M. , Gasatomizdat ~ 1963.
4. Sh.S.Nickolajshvili, "One--Velocity Problem on Neutron
Angular Distributions Emitted by the Point Isotropic
Source' Placed in the Centre of Sphere"~ Coll: "Theory
and Methods of calculation of a Reactor". M. , Gosatomis-
dat, 1961.
5. R.E.I{ynch, W.P.Jounson~ T.W.Beniot, C.D.Zerby ~ ORNI.-2292,
v. I-A.
6. V.G.Zolotuchin, S.M.Ermakov, Application oT Monte Carlo
method Tor calculation oT Nuclear Radiation Shielding.
Coll, "Problems oT Reactor Shielding Physics" edited by
Broder D.Z. and coll. D~., Gosatomizdat, 1963.
7. B.V.Gnedenko, A.N.golmogorov~ Ultimate Theorems Tor
Sums oT Independent Random variables. M. ~ Gosatomizdat
1948.
8. M.N.galos~ Nucl. Sci. and Eng., 16, No.1 III (1963 ).
9. S.M.Ermakov. Journal c~T Calculational Mathematics and
Mathematical Physicss v.39 No.1 186 (1963),
10. G.I.BEarchuk, V.V.Orlovy "On ad joint Tunetion Theory".
Coll. "Neutron Physics" edited by G.I.Marchuk~ Gosatom-
izdat~ M, 1961.
11. D.Zeliger, N.Iliesk~u, gim Ai Sang D.I,ongo ~ I,.B.Pickel-
ner 9 E, I. Sharapav. O IYT R-121 i3 ~ Dubna~ 196 3.
12, J.gennetts Z,M.Bollinger9 Nucl. Phys., 42, 249 (1959)0
13. E.R. Rae, B.R.Collins, B.B.Kinsey~ J.E.?{ynn, E.R.Wiblin,
Nucl. Phys. , 5 ~ 89 (1958).
1~.. M.NaNikolaev. Atomna~a Snergia, III 522 (1963).
303 _ 15
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