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Report written:

October 1955

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LOS ALAMOS SCIENTIFIC LABORATORY

of the

UNIVERSITY OF CALIFORNIA

LAMs-1969 This document consists of 24 pages

Q> ~

●

AU Los Alamos reports present the opinions of the author or authors and do not necessarily

reflect the views of the Los Alamos Scientific Laboratory. Furthermore, this LAMS report

is an informal document which has been prepared for a special purpose and is not considered

suitable for general distribution. It is accordingly requested that no distribution be made

without the permission of the Office of the Director of the Laboratory.

NUCLEAR PROPULSION

-+ -“; -”

.

-

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
.

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NUCLEAR PROPULSION

Report distribute~ ~~g 1 01956

AEC Classified Technical Library (H. E. Skinner, DMA)

AEC Classified Technical Library (D. J. Keirn, DRD)

University of California Radiation Laboratory, Livermore (Herbert York)

Rand Corporation (G. Safonov)

Manager SFO (Russell Ball)

Los Alamos Report Library

LAMS-1969

1-2

3-4

5

6

‘7

8-40

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● ✎ ✎☛✎ ● m. ● *. ● .. ● m

● O ● * ● 00: :C:ee

● **

● *8 : : : :0

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ABSTRACT

This report gives (1) a simple empirical equation

which should yield critical mass values of homogeneous,

partially moderated, unreflected spherical assemblies

with an accuracy of a factor two, and (2) critical mass

estimates obtained from (1) for Oy-C-H and OY-W-CH2

systems.

A

.

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.: ~-40: : :.” :: -. ● ☛ ● O* ●;O ●*; ●;. . . @liiuBEi “ ‘NcLh ~....-::- --

—.

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● ● *O

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● .9 :.. .

.

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i :iiiiiii

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9* . ● pQ_.—_= . .—-xii

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
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● . ● ● 00:0 ●: 8

● ** ● :*

9* :0 ● *

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CONTENTS

Abstract. . . . . . . . . . . . . .

- --

Page

3

1. A Simple Formula for Critical Masses of Partially

Moderated Oralloy Assemblies. . . . . . . 7

2. Critical Mass Estimates of Oy-C-H Systems . . . 14

3. Contribution of Resonance Capture to a = ~c/~f. 16

4. Critical Mass Estimates for 0Y02-W-CH2 Systems . 19

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APPROVED FOR PUBLIC RELEASE
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.__-... . . .. ~ .

killii- ● _.,

--------.;..9-

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
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● ✎ ● *9 9-9 ● 9*

● e ● 00: : .: ●0=0: ● a

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UNCLASSIFIED

1. A Simple Formula for Critical Masses of Partially

Moderated Oralloy Assemblies

It appears that critical mass estimates with an accuracy of a factor two are useful for orientation in the nuclear aspects of rocket reactors, the reactors of interest

1

containing anywhere from 2 to 27 kg of active material.

With this tolerance in mind, the object here is to give a

simple empirical equation for critical masses of spherical,

unreflected, homogeneous assemblies of Oy containing tnoderator and poison. By including reflector savings (typically,

a good reflector will reduce the bare Mc by a factor between

two and three), core shape, and inhomogeneities, one may then

proceed to critical mass estimates for the more specific

families of preliminary reactor designs.

The empirical critical mass formula proposed is

-3/2

Mc = p-1’2(U-235)

[

Xtr(v- 1 -a j

f(q) (1.1)

T=

n,

p(U-235) ● (l+cz)

● e .**.*. cc.. ● ● em

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::.:

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&ecal!n ● -o

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9*.

● ***

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(1.2)

UNC[A5JIF1ED

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
● m* ● ● e

4 9*9 ● ●am .

● 0

●

●

● :

●

● oe s9e—”oe O ● ee .e

● e,

: ● 0 ●0:: ● m :0 ● * ● m ●:0 : ●:. :*O ● 0

where

h!= = critical mass of U-235 in kilograms

p(U-235) = density of U-235 in grams per cubic centimeter

V= 2.5 = number of neutrons per fission

‘tr = macroscopic transport cross section of the U-235

diluent (moderator plus poison) averaged over the

fission spectrum

a = 2r/ X* = the ratio of macroscopic cross sections for

radiative or parasitic captures and fission

evaluated at thermal energy (except where

otherwise indicated)

f(q) = an empirical function graphed in Figure 1

& = slowing down power of moderator

A plausibility development of Equation 1.1 may go as

follows: Let the macroscopic cross sections properly averaged over neutron flux spectrum be designated o. The

buckling, B, of the flux distribution is then given by

B2 = 3(v-1-a)crfatr

Ignoring extrapolation lengths, the critical radius is

given by

Rc=;

Hence the critical mass is given by

4

[ 1

-3/2

MC=* p(oy) 3(V- 1 - a) Of rstr

3

(1.3)

(1.4)

(1.5)

‘he ‘alue ‘f ‘tr is very insensitive to spectral changes and

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● :Ai:.::

● *

iiiliiii ● :●

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
—

F

I“Jl=Jl! [ I

~

0 6

9* ● bb ● **

● 0

b ● *9

:0

● ●**

::..

.; ”:!””. ● O ● *O ● 9* ● *9 ● ** .*

● 0. & ● *O

● ●°:

● *O

● ea..

● *b*

●00..

● **

● *e . .

—o

F

0

N

0

>

APPROVED FOR PUBLIC RELEASE
● ✎ ●:e ● .* 8** . . . . .

● ● em ● 0

● *O* ● mmm

● : ● :

● e

: : ● m

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may be put equal to ~ tr. Barring poisons with epithermal

resonances, a also is insensitive to neutron spectra and

may be put equal to ~r/ ~f evaluated at thermal energy.

However, of is extremely sensitive to spectral changes and

we, therefore, place of = ~f g(~), where ~f = thermal

macroscopic fission cross section and g(~ ) is a function

of the spectral index ~ . We thus obtain, since ~f - p(Oy)

[

-3/2 ~T4

Mc=p -V2(oy) 1

Ztr(V- 1-a) ~ Lw7il ‘3’2(1.6)

The ratio of fast to thermal neutron flux is equal to

the rate at which neutrons are converted from fast to thermal

divided by the rate at which thermal neutrons are absorbed.

Qualitatively, the rate at which neutrons are thermalized is

proportional to the slowing down power of the medium, (2 s’

and hence, the ratio of fast to thermal neutron flux is

proportional to {~s/ ~f(l+a). ‘he parameter ~~s/~f(l+a)

is thus a spectral index, and indeed is an index for spectra

which have no thermal neutron content at all. Quantitatively,

a spectrum cannot, of course, be characterized by a single

index, CZs/ Xf(l+a), as details depend on, say, whether

the moderator is hydrogen, deuterium, carbon, etc. The

suitability of tZs/ Zf(l+a), or what is equivalent

● 9* ● ..* ● em . .

●°0 ● :

● a

:.l$;.::

● e

ii

● .iiiiii:’ ● me ● . ● ● *

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
b. ● ** 9*m .m8 ● m* ● a

● ,ae ● *

● *90b ●:**

● ***a

● a ● : :0

● e ●0: ●:0 9 ● ee ●

as a spectral index for critical mass estimates depends on

the existence of a unique g(~ ) in Equation 1.6 or better

yet a unique f(~ ) in Equation 1.1, since the latter may

absorb some of the changes with spectrum of a,

‘tr’ and

extrapolation length. The function f(~ ) was determined

from Safonovls multigroup critical mass calculations on

‘2°’ ‘2°’ ‘e’ and C moderated U-235 systems (RAND Corporation

Report R-259, January 1954). That is

1/2

[ 1

3/2

f(q) = Mc(Safonov) p Ztr(V- 1 - a)

The various f(~), ~ points obtained for the range

1 s (moderator/uranium mole ratio) < @

are plotted in Figure 1, and it is readily seen that, for

this range, the function f(~ ) is unique in so far as permitting critical mass estimates via Equation 1.1 to an accuracy within a factor two.

As the spectral index,~ , goes from 0.1 to 100, the

corresponding assembly spectrum goes from the typically fast

Godiva or Topsy-like spectrum to the thermal pile type spectrum. For ~ < 0.1, the standard one-group treatments for

$.

critical mass are capable of high accuracy not only because

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ikdNlbR ● ,*●

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● ***

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APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
appropriate

replacement

one-group cross sections

measurements (Los Alamos

Report LA-1708, July 1954), but also

are known, e.g., from

Scientific Laboratory

because of the possibility of normalization to critical mass values of known fast

assemblies such as Topsy, Godiva, and Jemima. The macroscopic

transport cross section, Ztr, appearing in Equation 1.1,

was defined as an average over a fast spectrum partly with

the hope that appropriate values could readily be obtained

from the atr tables of LA-1708. However, the high neutron

energy transport cross sections of Be, C, O, H, and D employed by Safonov in R-259 imply atr values considerably

higher than those given in LA-1708, and, in our construction

of f(~ ), average transport cross sections were based on

R-259 tables. A comparison is given in Table I.

TABLE I. ‘ALUES ‘F ‘tr

R-259*

Element (barns )

H 3.3

D 1.8

Be 3.3

c 3.0

0 3.1

*

Used for f(~ ) construction.

LA-1708

(barns )

z 2.0 (from LA-1525)

--

2.2

2.2

2.2

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
9**

● ●°:

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ebb

● ah

. . . ● .

9* ● 9* ● **

● *

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~mE: !“;

69 ●

● ** ● ** ● m ● *O ● *

Because of these differences, transport cross sections

for medium or large Z additives will be determined so as to

force agreement between critical mass predictions of Equation

1.1 and one-group treatments (based on LA-1708 cross sections)

for the limiting cases of no moderator.

A last remark concerns the quantity a = ~r/ ~f. This

quantity is of importance in severely poisoned assemblies.

Since the complete energy dependence of radiative capture

cross sections is seldom known, the value of a averaged over

an assembly spectrum can at best be guesstimated regardless

of how accurately the assembly spectrum is computed (with the

exception of the completely

measurements in Topsy Oy-Tu

proximate a values for fast

thermal system). Replacement

and Topsy Hydride can give apand intermediate assemblies,

respectively, and these, together with the known thermal

values, should be adequate w;ith the exception of poisons

having epithermal capture resonances. Equation 1.1 is invalid for systems having epithermal capture resonances if a

is evaluated from the above-mentioned sources. Validity can

be regained, however, if the epithermal capture is computed

and incorporated into a. The usefulness of Equation 1.1, of

course, depends on the simplicity and accuracy of this computation.

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
It is intended to estimate in this way critical masses

of the systems (a) Oy Cm Hn, and (b) Oy Wm (CH )

2 n“

2. Critical Mass Estimates of Oy-C-H Systems

These estimates, graphed in Figure 2, apply to normal

density, homogeneous, unreflected, spherical assemblies of

Oy Cm Hn mixtures, m and n denoting atomic ratios. Normal

density is here defined as that density which results from

additivity of volume of the mixture components. The normal

density, po, and cross sections of these components are

taken as:

po(oy = 93% U-235) = 18.5 g/cm3

po(c) = 1.67 g/cm3, ~as(c) = 0.74 barn, atr(c)

PO(H) = 0.173 g/cm3, ~as(H) = 20 barns, utr (H)

ac(oy)

OC(H) = 0.33 barn, UC(C) = 0.005 barn, =

Of (OY)

The values are calculated from the formula

= 3.0 barns

= 3.3 barns

0.20

[

-3/2

Mc(Oy) = 1.1 p-1’2(oy)

1

Xtr( ~- 1 - a) f(~)

&

v= 0093(l+a) p(Oy)

This formula differs slightly from that given in Section 1,

● oo ● ● ** ● ** ● 0

9** 9 ● **O

be : ● 0:0

● i4:” i :

9* ●:9 ● ** ● ** ● 99 ● *

!mw!ki

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
i 000

5

2

I00

.

10

5

2

I

● ☛☛

● ●☛✚

● *a*

040

● .*

8*e ● *

.99 ● m

9*

e

:

:* ●:* ●

r \- \ \ \ \

—

n=100

\

!

1 I I I I I I I I I I I I I I I 1 I

2 5 100 2 5 1000

FIG. 2. Bare sphere critical mass estimates of Oy Cm Hn

systems.

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
● ☛ ● ☛☛ 9*9 ● m. .0. ● .

9 **.*

baa. ● 9 ::

em :0 ● :

● ● :: :0 : ●:0 :.0.0

which applied to Mc (U-235) .

3. Contribution of Resonance Capture to a = xc/ Zf

It is desired here to develop (for use in treating

0y-W-CH2 systems) a simple expression for the contribution

of resonance capture to

Equation 1.1

Mc=p

[

-1/2 ~ ~

tr

Such an expression must

a and hence to critical mass via

1

-3/2

v- 1- a) f(~)

evidently be a function of the spectral

index ~ .

Since the average energy decrement factor f of customary

moderators is large compared to a

the flux equation in the vicinity

may be written approximately as

[

Zc(x) + Xs 1n(x) = Xs

where n+ is the flux density just

The resonance capture rate in the

r

capture resonance width,

of a resonance, x ~ s~n Ei,

n(x+~)=~sn+

above the resonance region.

ith resonance is thus

J J

xc (x)

(R.C.R.)i s ~ c(x) n(x) dx = ~s n+

Ec (x) +z~

i

●80●

●

●

● *

9*

● m

● 0

● 0

● *

● 9

i

● om ● ● ** ● 00 ● * ● 00::

● ● ● :* .*

.!.!:.i6:00 :.”

&!!8a!Q ● 00 ●

dx

(3.1)

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
iiii!w

.* .99 ● m. ● . . 9*9

● 00 ● O

.b. *m* ●

● ☛☛☛ ● ✚

● ●:0 : ●:0 :00

● 0

●

9

●

90

.

.9

● *

● 0

● *

● *

● 0

where the identically equal sign defines the effective resonance width, y. . The total resonance capture rate is then 1

(R.C.R.) = ~s n+ z ~ (3.2)

i Ei

Since the rate at which neutrons are slowed down past

the resonance region is equal to the rate of thermal absorption (strictly, the rate of absorption below the resonance --

a distinction we shall not belabor), one has

(3.3)

= ~a(thermal) ~ (thermal)

where #(thermal) isthe thermal flux. Eliminating n+ between

Equations 3.2 and 3.3, one has the standard expression

(ROC.R. ) s

[;::) Xa(thermal) # (thermal) (3.4)

The contribution to a due to these resonances is then

zf(average)+ (total)

(3.5)

Adopting for the moment a two-group notation, ~f(fast),

● *O ● ●**

●

● .: .:ol!!*

● 00 ● 00 ● *

● ● a

:0

● * ::

: ● *

9*9 :** ● *

● ** ● ●

0 s c **O

● ● 00 ●

●

; :.: ●

● a

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
,m . . . . . . . .

●

●

●

● e

●

● ☛ ● 0* ● ** ● 0. ● 00 ● *

● * ● , ● 00:: .: ●

● om :

: :0 :0

~f (thermal), etc., and””;e~~~~g “ ““” ●

X ~ (average) = ~ ~(thermal) g(~ ) (3.6)

where as usual ~ is the spectral index ~~s/p(l+a), one has

Xf (average) =

~ ~ (thermal) # (thermal) + ~ ~ (fast) # (fast)

# (total)

# (total) = # (thermal) + # (fast)

From this follows

# (thermal) 1

~ f (average) + (total) = ~f (thermal)

(3.7)

[

1- X f (fast)/ ~f (average)

1- ~f(fast)/ ~f (thermal) 1

Because of the normal limitations of two-group treatments, the

appropriate value of ~ f (fast ) depends on the assembly spectrum.

Indeed, as ~f(average) changes over the factor z400 from

thermal (~ = @ to fission spectrum (~ = O), X f(fast)

changes by approximately the factor 20, Thus

Xf(fast)

~

~f(thermal)

Ih30) g(q) (3.8)

and Equation 3.7 becomes

+( thermal) 1 l-~ g(o)/g(~ )

*

~f(average)~ (total) = ~f(thermal) 1 - llg(o)g(q )

rul- 4g(o)/g(q ) (3.9)

= ~f(thermal)

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
● O ● e*

● ** ●°0 : 980 :.wiimm ● *

...● om ● ** :99

● ● 9

● : :0 ● 9

● :0 ●:

● ●:* : ●:0 :00

● e

●

●

●

● e

.

.0

::

● 9

9*

● *

From the development of the critical mass formula, it follows

that

[13/2 f(q) ~ .-g(o)

f(o) g(~ )

so that, combining Equations 3.5, 3.9, and 3.10, one has

(3.10)

[) Aa2– 1X2 ~ + a(thermal)]

t

i. E.1 {+3”3}‘31

which is the desired approximate expression giving the contribution to a due to resonance capture in terms of the

resonance characteristics, yi/Ei, and the spectral index ~ .

If the resonances follow the Breit-Wigner relation with

peak ~. and width r, then

4. Critical Mass Estimates for Oy02-W-CH2 Systems

The results, graphed in Figures 3 and 4, apply to unreflected, homogeneous, normal density, spherical assemblies

Of [(O@2)xlV]yCH2, where x is the volume fraction of Oy02 in

the 0y02-W “fuel”, and y is the volume fraction of fuel in

the fuel-moderator composite. These results follow from the

“jam-handy” critical mass formula

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
● 0 ● 99

● ●*m : ●°0 ● .*:

mm 9.* ..* . .

B*9*

I

. 9

● 9 I I

● .

● a ● .O ● ** ● ●:9 ●

5 .

2 -

z—

o. F Mc ‘lTE —/— ‘c

5

2

I

.0 I

F

5

2

I

lNF

\

\

Km=l

FIN TE

.00 IL , -p-

.1

FIG.

X = VOLUME FRACTION lJ02 IN U02-w(=FuELl

3. Estimated M- = 00 contour for

● e 9** ● *9 ● ● ** ●

::0 ● m

● 0 .:; .:. .~ie!. ::: ● *

homogeneous

I

U02-W-CH2

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
E I

0

/

0

r0

0

I

! 0 I I

—

1 I I 1 1 1 I 1 1 I

In N m N

I

● m. ● ● ** ● 80 9* 9°0 ● 9* ● : : :0 ● m

● m

● ● .::”::

80 ● 00 9** ● *O ● 00 ● *

-—,

.h.. .-

APPROVED FOR PUBLIC RELEASE
● ✎ ● ☛☛ ● @m .Oom%-. o

● O

. .

● 00: ● : .: ●

● be ● :

● . :0 :.

● * ●0: ,00 ● ●:9 ●

[

-3/2

Mc = 1.1 p-1/2(oy) 1

Xtr(v-l-a) f(q)

and the nuclear data :

‘tr (CH2) = 9.6 barns

~ as(CH2) = 40.7 barns, P(CH2) = 0.8 g/cm3

otr(oYo2) = 11.4 barns

~as(Oy02) = 1.0 barn, P(Oy02) = 11.0 g/cm3

‘tr

(w) = 4.4 barns

f as(w) = o, p(w) = 18.8 g/cm3

@o/cJf(oY) = 0.077 + (~ - 0.084 ~2)/(3 + 2 ~2), excluding

resonance capture.

oc(oY)/af(oY) = 0.15 for I < 0.l

= 0.20 for ~ > 0.1

The Breit-Wigner resonance capture parameters used for

tungsten are:

E~es(ev) r (ev) ao(barns)

4.15 0.124 855

7.8 0.148 414

19.2 0.397 2810

These values of r and a. correspond approximately to the

room temperature Doppler-broadened lines. The nonresonance

capture cross sections in tungsten are not well known. The

● ☛ .*m ● ** ● ● m.

● ● ● 8 ● ● ●*.

90● 00:

● m

● m ..: ● ~.* A .:0::” ~

~2$””kiii

I

APPROVED FOR PUBLIC RELEASE

APPROVED FOR PUBLIC RELEASE
b* ●** O. **m ●

~wluBih= ‘“ ● m ● *. ● a9 ● 00 ● ea .*

● ● @*

● : ● * ● * ::

● a :0 ● * ●: ●:0 :

● 9

● ●:0 :00 ● 0

The empirical equation given on the preceding page is consistent with the following values:

Oc(wvaf(oy) = 0.077

= - 0.25

= 0.035

for Godiva (LA-1708)

for Topsy-Hydride (LA-1159, May

1950)

at thermal (Brookhaven National

Laboratory Report BNL-325~ July

1955 )

Since one is presumably interested in volume fractions of

U02 in W less than z 30%, it is clear from Figure 3 that

(aside from the U02-W metal system, i.e., y = 1) the fuelmoderator volume ratio is restricted to the range 0.006 to

0.08, of which Figure 4 covers only from y = 0.01 to 0.05.

In view of the uncertainty in nonresonance capture cross

sections of W and the very nonrigorous treatment of resonance

capture, one must certainly take the quantitative results in

Figures 3 and 4 with a grain of salt. As an example of how

the above uncer-tainties are reflected in critical mass estimates, we tabulate below the neutron fates for the

Y = 0.025, X = 0.25 case:

Fission in Oy02 40.0%

Radiative capture in Oy02 8.0%

Radiative capture in CH2 10.4$4

Nonresonance capture in W 14.3%

Resonance capture in W 11.4%

Remainder ~ Leakage 15. 9%

900 ● ● @* ● ** ● O ●°0● ;~~;:.;:

● .●--- 0-<<000_- ● 00 ● ** ● *

.. ~-:~-:. . . ●:*,

&

,---- *-ebee

● 6* ● a--,●

.W+*b● m

./

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(

● 0 ● *b ● ** ● O9 .ee ● *

● m

::*: : .:. .

● * ● ●

● . : : :*

● - ●*: ,9* ● ●:0 Q

The critical mass is related to the inverse 3/2 power of

the leakage probability so that a factor two uncertainty in

critical mass results from an z 60% uncertainty in leakage,

or in the present example a 30% uncertainty in total W

capture.

The tabulated numbers are somewhat artificial since the

first two percentages are essentially defined constants not

subject to error, the actual, or Safonov, spectral dependence

of ac(Oy)/af(Oy) being included in the f(~ ) calibration.

This artificiality is more than compensated for by the ease

with which replacement data on various poisons can be incorporated into Mc calculations; viz.,

ac(poison)/af(Oy) Z 1.3 ‘(poison)

AK(oy)

● 0 ● 9* ● ** ● ● *D

● ● 9 ● ● 9 ●°0

● m 900 ● 00:0

● **

● * •O~O.:_~.o ● 00 ● 0

$$ii!iiii

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