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Alpha-Like Calculations with MCNP

D. Kent Parsons, ESH-6

American Nuclear Society 1997 Winter Meeting, November 16-20, 1997, Albuquerque, NM

Los Alamos N A T I O N A L L A B O R A T O R Y

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Alpha-Like Calculations With MCNP

D. Kent Parsons

Introduction

Alpha ( t i m e - a b s o r p t i o n e i g e n v a l u e ) c a l c u l a t i o n s are n o t an

e x p l i c i t c a l c u l a t i o n a l o p t i o n i n MCNP1. N e v e r t h e l e s s , it i s

p o s s i b l e t o pe r fo rm a l p h a c a l c u l a t i o n s w i t h MCNP. Such

c a l c u l a t i o n s are p r e s e n t l y e i t h e r v e r y i n e f f i c i e n t o r r e q u i r e

special c o d i n g o r c r o s s s e c t i o n l i b r a r y m o d i f i c a t i o n s . However,

a l p h a - l i k e c a l c u l a t i o n s can e a s i l y be performed w i t h MCNP u s i n g

t h e KCODE o p t i o n w i t h n e u t r o n energy- o r t i m e - c u t o f f s . These

approx ima te a l p h a - l i k e c a l c u l a t i o n s are described and tes ted

below.

Alpha Calculations

The a l p h a e i g e n v a l u e i s d e f i n e d by t h e f o l l o w i n g e q u a t i o n ,

where t h e c o n s t a n t a can be viewed

2 ) CDdEdr = vZ,@dEdr

as t h e c o n c e n t r a t i o n

V s absorber r e q u i r e d t o b a l a n c e t h e p r o d u c t i o n o f n e u t r o n s

of a

( t h e

l / V

r i g h t - h a n d side) w i t h t h e loss o f n e u t r o n s ( t h e l e f t - h a n d s ide ) .

A t c r i t i ca l , a = 0 . 0 , t h e a l p h a s o l u t i o n and t h e more commonly

u s e d keff = 1 . 0 s o l u t i o n are i d e n t i c a l . I n s u p e r c r i t i c a l

s i t u a t i o n s , however, t h e a f l u x d i f f e r s from t h e keff f l u x

space and ene rgy .

i n

A direct c a l c u l a t i o n of a i s poss ib le i n MCNP by s t a r t i n g

n e u t r o n s i n a s u p e r c r i t i c a l sys t em and a l l o w i n g t h e n e u t r o n s t o

m u l t i p l y u n t i l some p r e d e t e r m i n e d problem t i m e t e r m i n a t i o n .

(Without t h e t e r m i n a t i o n , non-KCODE s u p e r c r i t i c a l MCNP

c a l c u l a t i o n s would n e v e r end! ) Ta l l i e s of t h e n e u t r o n p o p u l a t i o n

( f l u x divided by v e l o c i t y ) can be t a k e n w i t h r e s p e c t t o t i m e and

a d e t e r m i n e d by:

where N, and N, a re t h e n e u t r o n p o p u l a t i o n s a t t i m e s t, and t2. a

can be e v a l u a t e d o n l y a f t e r a s y m p t o t i c n e u t r o n p o p u l a t i o n

b e h a v i o r h a s been e s t a b l i s h e d and t h i s can r e q u i r e i n o r d i n a t e

amounts of computer t i m e .

An a l t e r n a t i v e approach2t3, ana logous t o t h e a l p h a s e a r c h

a l g o r i t h m s i n d e t e r m i n i s t i c codes, i s t o add an everywhere-

p r e s e n t l / v a b s o r b e r t o a s u p e r c r i t i c a l KCODE problem and

c a l c u l a t e keff . The " c o n c e n t r a t i o n " ( i . e . , t h e m u l t i p l i e r a) of

t h e l / v a b s o r b e r i s t h e n a d j u s t e d till t h e code c a l c u l a t e s a

modified keff = 1 . 0 . T h i s p r o c e d u r e i s n o t ava i lab le i n MCNP.

N e i t h e r i s t h e r e a l / v absorber avai lable i n t h e c o n t i n u o u s

ene rgy c r o s s s e c t i o n l i b r a r i e s .

A p l a u s i b l e workaround f o r t h e r m a l o r e p i t h e r m a l sys tems i s

t o u s e a p p r o p r i a t e l y no rma l i zed c o n c e n t r a t i o n s of t h e l / v

absorbers He-3 o r B-10, b u t t h i s p r o c e d u r e f a i l s i n f a s t sys tems

due t o t h e non- l /v behavior of H e - 3 and B-10 a t f i s s i o n spec t rum

e n e r g i e s .

Alpha-Like Calculations

For approx ima te a l p h a - l i k e c a l c u l a t i o n s , t h e normal KCODE

o p t i o n o f MCNP i s u s e d i n c o n j u n c t i o n w i t h a t i m e - o r an energy-

c u t o f f (see t h e c u t : n ca rd ) . An ana logous p rocedure i n space i s

t o t r u n c a t e t h e ou te rmos t r e g i o n s of t h e r e f l e c t o r o f a

s u p e r c r i t i c a l sys tem. I n a l l t h r e e cases ( o r i n any combina t ion

of t h e t h r e e ) , a d d i t i o n a l n e u t r o n l o s s e s a re b e i n g added t o a

s u p e r c r i t i c a l sys t em t o f o r c e b a l a n c e between p r o d u c t i o n and

l o s s e s . The t i m e - c u t o f f adds something s imi la r t o a b s o r p t i o n ,

w h i l e e n e r g y - c u t o f f s have t h e effect o f an everywhere-present

n e u t r o n a b s o r b e r which i s t r a n s p a r e n t above t h e c u t o f f , b u t b l a c k

below t h e c u t o f f . R e f l e c t o r t r u n c a t i o n adds e x t r a l e a k a g e -

which i s a l o c a l i z e d effect ( r a t h e r t h a n everywhere-present ) i n

an o p t i c a l l y t h i c k sys tem.

S i m i l a r t o d e t e r m i n i s t i c a s e a r c h e s , t h e t i m e - o r energy-

c u t o f f s o r t h e r e f l e c t o r t r u n c a t i o n are varied t o f o r c e t h e

mod i f i ed keff t o 1 . 0 . I n fact , o n l y t h e t ime-cu to f f p r o c e d u r e i s

un ique t o Monte C a r l o methods.

There i s p h y s i c a l m o t i v a t i o n f o r a l l t h r e e p o s s i b l e c u t o f f s .

I n a rapid n e u t r o n i c t r a n s i e n t , any n e u t r o n which h a s t o o l o n g a

l i f e s p a n ( t i m e - c u t o f f ) w i l l be " l e f t behind" . S i m i l a r l y , any

n e u t r o n t h a t d r o p s t o t o o low an energy ( e n e r g y - c u t o f f ) o r

wanders t o o f a r o u t i n t h e r e f l e c t o r ( r e f l e c t o r t r u n c a t i o n ) w i l l

a l s o be l e f t b e h i n d .

All t h r e e c u t o f f p r o c e d u r e s can be viewed as s u b s t i t u t e s f o r

the everywhere-present a/v absorber. To the extent that they

approximate the neutronic effects of that a/v absorber, the

modified KCODE calculated flux distributions will approximate the

true a flux distribution.

kef*

a

Sample Results

1.062 0.001

0.043 sh-'

Sample problem 11 of a set of problems originally defined

for neutron lifetime calculations4 was chosen as a calculational

exercise. It is a simplistic problem with a 7 cm sphere of

highly enriched uranium metal surrounded by 10 cm of graphite.

MCNP4B results for sample problem 11 for each of the three

11 reflector

cutoff

figure

energy

5.65 cm

methods are presented on the table below as well as the

of merit (on the combined estimate of kef,) and the average

of the a/v (or equivalent cutoff) absorptions.

Type of

Calculation

Result

time-cutoff ll 6.55 sh

energy-cutoff ll 51 keV

11 truncation

Figure of

Merit

4112

4701

5370

5358

6894

Avg. Energy of

l/v Absorption

430 keV

63 keV

-51 keV

-1345 keV

. Fission source distributions for each of the 5 cases are

plotted on the figure. The similarity of the approximate alpha-

like fission source distributions to the a distribution is

readily apparent, as well as the significant difference from the

keff distribution.

1.2

1.1

1

0.9

0.7

0.6

0.5

I 0.4

Fission Source Density Across the Spherical Core

l I I I I I I I I , I ~ ( J , l ( J I I l l l l l l , l

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

+ keff --if- energy-cutoff

-t- alpha -f- time-cutoff ~. 0 ~. reflector truncation

Conclusions

MCNP KCODE calculations with time- or energy-cutoffs, or

with reflector truncation are similar to a calculations. These

alpha-like calculations give spatial flux distributions which

resemble true a distributions. Unfortunately, the equivalence is

,

not exact due to the failure of the various cut-off mechanisms to

match exactly the l/v energy dependance of the a/v absorber.

. ~

References

1. J u d i t h F . Briesmeister, E d i t o r , "MCNPTM--A Genera l Monte

Carlo N - P a r t i c l e T r a n s p o r t Code", LA-12625-M, Vers ion 4B, March

1997.

2 . Dennis Brockway, P a t Soran , and P a u l Whalen, "Monte-Carlo

E i g e n v a l u e C a l c u l a t i o n , " i n L e c t u r e Notes i n Phys ic s , V o l . 240,

Monte-Carlo Methods i n N e u t r o n i c s , Pho ton ic s , and S t a t i s t i c a l

P h y s i c s , P r o c e e d i n g s of t h e J o i n t Los Alamos N a t i o n a l L a b o r a t o r y

- Commissariat a 1 ' E e n r g i e Atomique Meeting a t Cadarache Cast le ,

Provence , F rance , 1985, pp. 378-387, o r see LA-UR-85-1224, L o s

A l a m o s N a t i o n a l Labora to ry , 1985.

3. W . Goad a n d R . Johnson, "A Monte Carlo Method f o r

C r i t i c a l i t y P r o b l e m s " , Nuc lea r S c i e n c e and Eng inee r ing , V o l . 5,

pp. 371-375, 1959.

4 . John Hendr icks , Kent Pa r sons , Greg Spr iggs , and Bob Busch,

"Comparison of Neut ron L i f e t i m e s as Predicted by MCNP and

DANTSYS", T r a n s a c t i o n s of t h e Amercian Nuclear S o c i e t y , V o l . 76,

pp. 365-367, J u n e 1997.

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