EFFECTIVE CADMIUM CUTOFF ENERGIES

. I

ORN L-2823

UC-4

- Chemistry-Genera

EFFECTIVE CADMIUM CUTOFF ENERGIES

R, W, Stoughton J. Halperin Mo P o Bietske

.

U N I O N CARBIDE CORPORATION for the

U.S. A T O M I C ENERGY C O M M I S S I O N

.

1

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1

om2823

DATE ISSUED

OAK RIDGE PTATIOTJAL U B O W O R P

Oak Ridge, Tennessee operated by UT?IOIT CAR3ZDE CORPO€U!ITOT? f o r the Ud S + AT01!IC EIERGY COPdNISSION '

8

3 4 4 5 6 0363434 0

-2

-

Effective CadmLum CytofZ Ehergies R . W . Stoughton, J. Halperir,, and Marjorie P, Lietzke

Abstract Effective cutoff energies for poir-t

_1.I/. absorbers

inside of spherical

and cylindrical cadmim f i l t e r s have been calculated f o r thermal reactor neutrons. plus a

The neutron s p e c t m was assumed t o consist of a Maxwellian

1/E component,

and the parmeters varied were the thickness of

f i l t e r , the Maxwellian tenrperature and the b ~ w e l l i a aLo

_1/E flux r a t i o .

Because of the s e n s i t i v i t y of the el’fective cutoff t o Maxwellian f l u x parameters for t h i n f i l t e r s it i s recornended %‘fiatf i l t e r thicknesses of about 40 m i l s be used.

Forty mil f i l t e r s show effective cutoffs a t about

0.50 t o 0.55 ev. for temperahres up t o about 500°A ( o r about 0.045 ev).

Effective cutoff energies f o r boron f i l t e r s were also calculated for purposes of comparison.

The cutoffs f o r cylindrical cadnium f i l t e r s

should be applicable to a properly designed experLxen+ual f a c i l i t y .

-3h t r o d u c t ion

For many yems cadm3m has been used as a fiJ,ter for thermal neutrana in maswing t h e r m 1 vs epithermal reaction r a t e s f o r reactor spectrum neutrons. Yaa e f ' f e c t l v s Cd cutoff er,ergy kras been estimateil as several

depenaing on the t h i c h e s s ~f the f j l t e r .

Hughes

a

e s t h a t e d about 0.4Q and

0.57 ev for 10 and 30 m i l cadmium -t;taFcknesses, -respe:tiveljr, calculated r e l a t i v e number of neufxoos transmi3Aed vs enezgy Dayton and Pettus

tenths ef an ev

from a p l o t of %he

.

More recently

calculated effective Cd cutoff energies f o r flat f i l t e r s

of ,large area f o r both a monodirectional and an isotropic flux. only a

& flux,

w i t h lower wii upper lj.aits of 0.025 ev

%he upper liiit was not

and energy.

They assumed

aQa 10nev (althorn

important) i n the* numerical integrations over the angle

They defined the effective cutoff energy as the cutoff of a perfect

( I n f i n i t e l y sharp) f i l t e r that allows t h e same tu"al xmiber of-absorptions by the f i l t e r e d sample as does the gLven cailmkm ffl%er. In - h e i r calczila%irms they

assumed that the f i l t e r e d sanple had a The purpose of thLs paper is

tb

& cross

sectian.

calculate ef"fec-i;5.-veCd cutoff enezgies, as absorber and for an idealized

defined by Dayton and Pettus, f o r a 12ypo-khetical&l

Two geometrical configurations have been conside;-ed:

spherica4. and cylLndrica1

f i l t e r s (one cm dim. by two cn high.) ix resctor ixz*adiai;ions. Fdihfle %e ampl.es

occurred) and rather smal.1 3x1 ares,.

Xt c a s be s$mm that if a point san2le is moved

a signtficant distance f r D m the cen%er of the cyl-bdrical s h e l l i t s correaponfiir&

effective C d thickness is changed very l i t t l e .

For example, i f t h e p0b-k is moved

along the axis half way t o ogle of %he f a c e s (i. .e.

to one q w t e r '3f t h e height from

the nearer face) the effective Cd thickness is changed about G.$.

Eence a th4.n

-4f i n i t e sample which approximates an i n t e p a l of such points should shov very rxarly the sane effective cutoff as does the central point.

Actually an e f -

fective thickness could be calculated and a correction made, i f desired. the case of the spherical f i l t e r s the correction would be larger.

In

From an

experimental point of view, however, the cylindrical f i l t e r s are easier to make and to use, i n addition L ' o allowing an effective cutoff which is r e l atively insensitive t o position of sample. Effective cutoff energies were also calculated f o r boron f i l t e r s LEIorder t o compare the variation of effective cutoffs with f i l t e r thickness and spectral parameters t o t h a t of cadmium.

I n general r e a l fiYcers such as caclnriurn a.~ad boron o n l y approxiaate o m defined perfect f i l t e r .

An additional factor t h a t a f f e c t s the u s e m e s s of a

f i l t e r is the sharpness w i t h iflich the t r t z x i t f o n fi-on opacity t o transparency i s made a t the cutoff.

A siaple measure of t b . i s efficiency was made here by

calculating the f'raction of %he ",tal number of reactions occurring i n t h e l / v t e s t sample a t energies below <&e ccnpiAted cutoff energies (FRACT).

hypothetical perfect f i l t e r would yield a m C T

= 0.

TIIUS

s%e

A comparison of t h i s e f -

ficiency f o r cadmium and boron f i l t e r s has been made i n this paper.

In general

much lower values of t h i s pmameter FRACT were found for cadmium than f o r boron. This i s of course consisten5 w i t h the large resonance b cadmium giving r i s e t o a sharp cutoff L a contrast t o %he 1/77 e r m s section dependence of boros.

Model, Assmqtions a d Calculations The f l u x was assumed t o be isotropic w i t h respect t o the point &I absorber and t o consist of a Mamellian thermal plus a spectrum.

Following Westcott

a

the

1/E:f l u x was

1/E e p i t h e m l

assumed to go t o zero

(sharply) a t five times the Max~relliancharacteristic energy E (where na Em = E ) . For a Maxwellian spectnm the Cerivative of the neutron density n I

-

,

-5-

-

with respect t o the energy E is given by the equation

-

-

,

The d i f f e r e n t i a l of the t o t a l flux Ftot in any interval dE then becanes

Ibr is the &pithermalor flux per u n i t E .On changing the vasiable to x e E/Em, the d i f f e r e n t i a l of the total reaction r a t e R pes atom f o r a l/v -

where

I &

I

absorber hecomes

where

#r = o far xC5 ( i * e . $ E < SE,). Here the subscript zero r e f e r s to a standard energy or velocity (here taken 2200 m/s).

.

where r 5 nvo/pI,B i,e,, the r a t i o of the "conventioW" t h e r m a l flux

t o the resoname flux per unit l d 3 ,

a

as3

-6Two geometrical configurations w u l b e considered8 c a l cadmitrm

shells with p o b t

qheriea3. and c y W i -

absorbers a t t h e i r cen%eps.

the only neutxom going through the spherical s h e l l s

A@ $ c a faa F i g o

L

ax% a&wgoing fJTpougbt h e

point saqples a t the centers go &Low a diameter, and hence %he t\haabes~of cadmium traversed before reaching tihe ssanples is just %he act;;lsal thickness of %-

the spheres lo, ETfects of scattering of neutrons by the &heJJ.f!& wD"M mt h

-

general be important and EtSe not coruidered h these c*%-,

-

same definition_,& effective Cd autoff energy

Pettius

a

Using the

Ec as tha.1; w e d by Day-bn and

the equation f o r the spheres becomes

I

where

zcdlois the macroscopic cross section of

cadm;lum titxts

the t;hbckness of

where x' = x C

6

if

X@

p 5

J

-7-

UNCLASSIFIED ORNL-L R-DWG. 38 132.A

Cy Ii nder

Sphere

A

D eL = t o n - l H

H

Fig. 1 .

Spherical and Cylindrical Cadmium Filters.

-8

-

P

Eqwtion A?) was uSed t o evaluate t h i s integral f o r va.lues of xc greater tfiaJz 5;.

For d u e s of xc l e s s than 5$ the f i r s t integral on the l e f t w a s divided i n t o

-

two parts:

that from xc t o 5 and that above 53 the first part was evaluated

-

numerically using Gaussian quadrature and equation (7) was used f o r the r e mainder of the integral. The magnitude of the error made i n u s i n g equation (7) is less than the

last term i n pasentheses. .wlan O,@

For values of xc greater than 5 this error is l e s s

-

It i s not recommended that t h i s approximation be used for xc values

-

l e s s than about 5.

The two integrals on the right were evaluated numericaw using Gaussian qpa.cbature.

L.1 the numerical integration f i n i t e upper lWts had t o be set, so

- = 1000.

all the four upper limits i n equation (5) were s e t a t x

It can be shown

a t the error made in setting these lwts a t 1000 i s l e s s t h a n about Od.$ in the reaction r a t e w i t h even the thicker cadmium shells, by expanding the exponential in the last integral, dropping a l l terms a f t e r the second and integrating ana3-ytkallyo It should be noted t h a t the error i s not as great as t h e values

of the integrals above x = 1000, since these approximately cancel on the two

sides of equation ( 6 ) .

-

After evaPuat%ng the right haad side of ( 6 ) , xc (and hence Ec) was e v a L

-

uated by successive approximationso The ORACLE (the Oak R i d g e National Lab..

oratory d i g i t a l coqpvter) and the bBM-704 a t the Oak Ridge Gaseous Diffusion P l a t were used for ~ e s calculations, e

Also calculated ia mimy ca6es was %he fraction

FRAd of tota3 reactisns

of %.he central l / v sanple which occurred below -tihe effective anitaff. The Torn of the expression used for the cmas sectFon of Qa was a one

l e v e l Ere2totligner b m parameter equation in whiah the parameters w e r e a& juated ta fit the abawption curve given in ENL-325

u*

Fkplicit2.y %he

equation w a s

5.076 x

2&lQ=

section of

lo

fi[4(~&0.178)~ + O e 0 l 3 ~ ]

-

where lo 2s in m i l s of metallic Cd and E is i n ev.

!&e

t o t a l absorptfon woss

cadmium if3 essentially due t o -&he 0.178 ev rescmauce of Cdu3,

at

l e a s t out la an energy of mqy eve Hence the paratnetera In equation (8) axe those for this nuclkle a t its wbural i s o t q i c abundanoe.

I n the case of the cyljtndrica,l shells the -tzraa&ssion seen i n Fig. 1, depwds -on

of p 3 ~ u t r o ~as ,

the angle betiieen the &ireation of t h e neutron

aad t h e axis of the c y l b d e r as w e l l as upon the neutmn energy.

In eqw-tion

-

-

(8) lo wt be replaced by lo/cosO and lQ/sin@f o r values of 8 be%ween zero

-

a,nd

eL

=

taS-l(D/H) and between the latter value and

-H and D- m e %heheimt and dianeter, changing the variable cb & ..s cosB2

& respecd;ivelye

respectively, of %he cyl-ers.

wing appwxbation (7) aad addigg

",

1

\

Sere

AfLer

-1G 2x1/2/ n

v m t o both sides,

the equation analogous to

(6) b e c d s

Here the upper l i m l t of energy i n the numerical integration W a s taken as

-x = 1000 with the energy range being divided

into

15-100, 100-1000) and the range of 8 into 2 (i.ee, 0

tan-l(D/H) 3c

-

/

3t/2).

*

5 intervals (i.e

Dy

- tan*'(D/H)

0-2, 2 0 5 ~

a3ad

The fratemation was carried out using Qqmian quadratwe

Some calculations ware aJ-s.0 carried out using 10 energy intervals and these se&ts agreed t o w i t h F n OOl$ With those us5 intervals,

-

with 8 points per interval (of either x or

E)

on the

IEM-704 c q p u t e r a t the

O a k Ridge Gaseous Diffusion Plan”c.

Also calauJ.a,ted was the fraction F C T of the t o t a l reactions of the

l/v

sample which occurred below %e effective cutoff.

It should be noted that if the value of xc is Less than 5 the second ex*

p e s s i o n on the l e f t of eqwtion (9) should not be given

Fn the s e r i e s expansion

shown. Rather it shovld be evalwted, as in the sphere case, by

using the

following expression

The v a u e of xc

rimy

then be obtained by successive a p p r o x ~ t i c x susing

numerical integration.

This procedure was necessary i n Che ca3,aulatians on

both the thinner cadmlton a,nd boron f i l t e r s .

-

In the case of boron filters the energy range (values of x) w a s divided

into 8 i n t R r V a l s ( @2j 2-53 5.115; 15-Zoo j 1OOalOOOj 1OO0-10,000~ 1O,OOO+OO,OOO

j

100,000~1,000,000) because of the slow decrease of the boron cross section

w i t h energy,

The q q e r limits on aJ.l integrals here beccrmes 1,000,000 and

the t e r m added t o Mi21 sides becomes X;/~/~OO. the expression

xcdl -was replaced by = 0.042039

-

/’

where

-’

% x0

For %he boron cangutatkas

9

z & is the macmscopic aross section of bomp tlmes -

5 being i n the units of mg B/m2, 755 2.

tihe thicbpessp

The 2200 m/s m s s section used here was .--

Results azxd Discussion Cd @hereso Effective cutoff energies f o r spherical cadmium sheads,

are

presented in tabular form in Append& A and axe shown for three d u e s of the MaxweUian t o e p i t h e d flux r a t i o ;

as a function of the characteristic

MaxweUan energy Em = kT -in Fig, 2,

Values axe also giver, f o r

5 6

trans-

mission ( i.eo, the energy a t which the exponential tmnsmission factor is

-

equal t o ofie-ha3f) and f o r r = o (E, = 0,0253 ev).

It can be seen that tihe

cutoff energies a t 20 m i l s thiakness arid helow are sensitive

r as w e l l as E&,

even a t lawer values of the latter.

For 30

Iu

flE1ters the cutoffs are nearly independent of atwes,

5 a t the lawer

Die values f o r 5% transmission axe rather close t o

f o r the tbicker f i l t e r s and Lower t m e r a t w e s ,

1% would be desirable t o have a cutoff neas the assumed lower U.&t I & Plwc, S.e,,

tihe

Of

sm.However, this would require thin filters, which-,aPe very flux r a t i o 2, Xence the authors suggest that the best ccnnpro-

sensitive -bo the

mise wo7fld be achieved wit21 f i l t e r s of about 40 m i l thickness. Approxiasbe calculations for cyEin&ical and cubic f i l t e r s were &e,

based

on the sphere calculattons aml the a s s q t i o n that the transmhsion could be calculated sinlply by using effective thicknesses seen by the neutrons 3.n the

+aansmission ex;ponentiizl factor,

"he effective thickness f o r cylinders cari

e a s i l y be shown t o be lcL3 and L,22 times the actual oihickness f o r an (height/dianeter) of 1 and 2, respectively.

Then the appro-te

a

effectAve

cutoffs for a cyLLnder w o u l d be given by the r e s u l t s for a sphere w i t b a thickness of 1.13 and l,22, respectively, tjmes t h e a c t u thickness of t h e cylinders,

Such approxiw,te and accurate cutoffs f o r a a@dader

(H/D = 2)

UNCLASSIFIED ORNL-LR -DWG. 38139A

.80 mils Gd for 50% trans

.70

- 120

nils Cd for r = O

& =.025; 120

120

100

400

ao

80

60

60 50

400

.60

f

.50

-

ao

aJ

lu" .40

50

50

40

40

30

30 ut.z.=L._L.L.L.

- 20

10

.30

-r=20 --- = 30

,-

".&.-...... _ ..... - ........... ... &.-...... --........................ ...... ................................. ......... -----------_______ ............ --- -_ ............. .......... -- ............ ................................ ..................................... -----__________----- -----...... ............... + .......... ............ ............................................................ L:

-- --. ..--. -_ -. -. --

20 15

-----_ -----__ -. . --. .---. --.. --- -----.......-..-..--.......... -----___ -_____________-_ --------........................................................... --.---_________________--------------L....

.L

Y'.L"'"t".

30

Y

............

e..

60

h

;"e....

e..

- 40

>

--

'V

...................... 20

10

-

I

...........................

A

w

- __-- ----_ _ _ _ _ _ _ _ _ _ _ _ - - ----------- ............................. _---_----- - .................-2-:*: .............................. .......
I

_/c*

20

-

=

Rperfec+ filter

.IO

.(

>-

.O3 294

Fig. 2.

.04

.O5

.06

580 MAXWELLIAN TEMPERATURE

filter

(E,) for

.O8

.O 7

870 (OA)

Effective Cadmium Cutoffs for Spherical Shells.

l/V

absorbe .09

3

4160

are given i n Table 1 f o r two values of Em (0.0253 and 0.05 ev)

-

.

Here the

f i r s t column gives ?he actual thickness, the second gives the effective

thickness and the last column gives the energy a t which 50% of the neutrons a r e transmitted.

The corresponding thickness r a t i o (effective/actual) f o r

a cubic f i l t e r i s about 1.20.

Some calculations were also carried out under conditions which should have given r e s u l t s i n agreement with those of Dayton and Pettus

12). Using

-

the sphere model, described above, with no Mxwellian ( i . e . , r = 0 ) and with a lower l i m i t of the I & flux a t 0.025 ev the r e s u l t s should have agreed with

t h e i r monodirectional flux calculations f o r very t h i n samples (see the t h i r d and fourth columns of Table 11). The data of Dayton and Pettus were read from

*.

a graph but should be accurate enough t o indicate a d i s t i n c t difference

Their

resonance parameters f o r Cd were s l i g h t l y different from those used i n t h i s paper, but hardly enough t o account f o r t h e difference i n results.

-

-

Reactior, rates per unit energy vs E and t o t a l reactions below E have been calculated f o r the

l/v absorber

as a function of energy for 10 and 40 m i l cad-

-

The solid curves i n Figs. 3 and 3a show

mium filters, Em = 0.0253 and r = 20.

-

-

the fractions of the t o t a l number of reactions occurring a t energy E per unit energy; the dotted curves show the fractions of the t o t a l reactions which

-

occur between energies zero and E.

*A

The t o t a l number of reactions occurring

recent comunication ( 2 June 1959) from W . G . Pettus contained some recal-

culated values and these ( i n the case of the beam flux) were much closer t o the r e s u l t s obtained in t h i s work than were Dayton's and Pettus' original values.

(See Table I1 and r e f . ( 2 ) ) .

-15-

Table 9

Approximate vs Accurate Effective Cadmium Cu-toffs for Cylinrters

( a b . = 2) f o r Em = .0253 lo

Eff 0

sphere

cylixder

1

cake

10

122

20

24,4

30

36.6

40

48.8

60

732

100

122.0

approx.

-P = 20

-16-

Table I1 Effective Cd Cutoff Energies for Monodirectional

1/E F l u

F l u Lower Limit in ev

Cd Thickness (mils )

Dayton and Pettusa

This Paper

5 x

0.0253

0.025

0.025 -

10

0.362

0 -237

0.26

20

.431

.429

.34

40

,520

.520

.42

60

.584

583

47

100

.679

679

56

a

V a l u e s read from a graph

m.

- 17U N C L A S S I FlED O R N L - L R - DWG. 39143

-

T

I

I. 3

T

1.2

1.1

1.0 lu 3.9

6

cr W

0.8

z

3 0.7

u)

c

.o $ t

L

2Q.'

0.6

X-

s

cn z 0.6 0 -I o

1

IC

a

0.5

E LL

0

0.4 z 0 -

-Reaction

I I I

I

I I

I

I

Rate --- Fractions of Reactions

/o = 10 mil

-Reaction Rate ---Fractions of Reactions

/o =40mil

o

0.3

2 LL

0.2

0.1

I I

0.0

I-

A

I111111

lo2

I(

I I111111

103

I I1

io4

E,

(40 mil)

Fig. 3. Reaction Rates and Fractions of Total Reactions Below Energy E for Spherical Cadmium Shells.

- 18-

I

l1111l1

I

I

I I I1111

I I I IIll

I

UNCLASSIFIED ORNL- L R - DWG. 38934 I 1 1 l 1 1 1 I I I I I III

-

-

-

-

10’

IO0

l‘ > W

.-c u)

t

[r

3

z

W

W

h

L

lo-’ 3

W

L

13w

t .-

a L

m m

0

I I I I I I I

z

10-2

0

I-

o

a w

I

I

I I

[r

I

LL

0

IO-^ z 0

I-

i

0

a

I I

I

-

= 40 mil

---

= 10 mil

I I IIII

T

E

io0

I

I IIIIII

10

I

I1l111

40

I (40ECmil ) EC

(10 mil )

Fig. 3a. Reaction Rates and Fractions of Total Reactions Below Energy Cadmium Shells.

E for Spherical

a t all energies was noxmaLized to uni"cyo Acreactions in the 10 than tn the 40 mEL caae,

of course tihere m e more As seen

i n F i g e 3 about 6 ~ of $

the total reactions occur belaw the cutoff wit& Maxwellian neutrons I n the

BI the 40 mil. case very few reactions inm~,ve

case of the 10 mil khicknesh.

and only a few percent of t h e reactions w c u r be3.0w6EG.

MaxweUan neu-ns

-

The c a l c U t e d r e s u l t s f o r boron spheres m e presented

Boron Spheres.

i n Appendix B. Cd Cylinders.

The results for the cylindrical shells are Shown fn Fig.

4

4

The cutoffs are seen to be sfmilas t o but a l i t t k e higher than those for the spherical f i l t e r s , as expected.

Again a thickneas of abaut 40 m i l s is indi-

cated in arder that We cutoffs be insensitive to We flux r a t i o a t the lower Maxweuian teaperatures.

Since the melting point of cadmtum mtal i s 321%

0

(594 A) the higher temperatures ~ R be Y unrealistic; however cadmpum i n some chemical form could be used as a f i l t e r a t higher temperatures by c l a d d b g Vdues of E

it w i t h sume high melting material.

-

in Fig. 5 f o r r =

G

VS.

C d chiokness are

sWm

0

0,

E and

30 for

% -

= 0,0253 ev.

The fractions of the total reactions by the

& sample below the

effective I

cutaff are shown i n F&g, 6.

-

ing Em a t an;y thickness. creases wi-bh;

For r $

A t low

'c

E;m

0,

this fraction is increased by Fncreas-

( i . e e 2 mound 0,0253 ev), the fkaction in-

*-

only f o r t h i n f i l t e r s ; a t the higher Em values the fraction in-

creases w i t h r f o r all thicknesses.

A t En = 0.075 ev a dfsfinct rnihbtum

OCC'UU"~

d

in the fraction curve at about 15 mjls Cd,

A t Ea = 0.1 ev the mFn39aum &s broader -..-

and l&s d k t j h c t , See Appendix C f o r calculated values for cylindrical. Cd fil%wse

- 20 -

UNCLASSIFIED O R N L - L R - DWG. 381408

.8

.7

nils Cd

........ r = 12

00

-

............

60

.6

-

..................................

40

-T

..................................... ..*""*: _---------_

.5 30

>

............................

Q,

.......................................................................................... 20 -lu" .4 - - - - _ _ ---_---__ -- _ _ _ _ _ _--------------

v

. 3 -1 5

.2 10 .02

.03

294O

Fig,

.O 4

.O 5

.O 6 E,(ev)

-

.07

.08

580" 870° MAX WELL1AN TEMPERATURE ( O A )

.09

.I 0

1160"

4. Effective Cadmium Cutoffs for Cylindrical Filters (Height/Diameter = 2).

-21

-

- 22 0.L

0.:

z

0 -

I-

o 0.2

a

LT

LL

0.4

\&-,

I-= 0

20

40 mils Cd

60

=.0253 ev

80

400

Fig. 6. Fractions of Reactions Below Cadmium Cutoff for Cylindrical Filters (Height/Diameter = 2).

Boron Cylinders. VS.

-

Figs. 7a and 7 b show typical curvEs for Ec and FRACE

thickness, respectively, f o r boron and cadmium. As expectad, there is

as expected the boron shows a much W g e r FRAW

no leveling off of Ec w i t h increasing thickness i n the case of Boron.

Also

.wlan cadnlium azld shows t b e

effects of the Ma%wellim neutrons over a wider range of thioknesses.Also see Appendix D. Conclysiow Effective cadpl;lwn cutoff energies have been calculated for a wide va;riety of Maxwellian neutron temperatures, r a t i o s of Maxwellian to epithe-1 and thicknesses of Cd f i l t e r s .

flux,

Beaause of the sensitivity t o Maxwellian flux

parameters for thin filters (less than about 30 m;uS) it is recanmended that f i l t e r s of about 40 mils be used i n Cd r a t i o work.

Depedlng on the exact

conditions the effective cutoffs for 40 mil C d l i e between 0.50 and 0.55 ev f o r MaxweUm texperatures of 300 t o

5008.

The vaLues presented here Shou3.d

hold t o a good approxirpation for s m a l l samples centrally located inside cyl i n d r i c a l cadmium f i l t e r s .

I " I

+,

-

- 24 -

UNGLASSI FI ED ORNL-LR-DWG. 3 8 1 3 8 A

-B

---- Cd

6-

-

5-

T

4-

>

a, v

-

lu” 3 I I

I

-

2I

I -

-

‘-___________--------------.*/’

,

I

1

I

10 20 30 40 50 60 70 80 90 400

I

I I 10 20 30 40 50 60 70 80 90 100

mils Cd 4

I

Fig. 7a. Effective Cutoffs for Cadmium and Boron Filters for E = 0.0253 ev and r m = 20.

Fig. 7b. Fractions of Reactions Below Effective Cutoffs for Boron and Cadmium Filters for E = 0.0253 ev and r = 20. m

-25-

Acknowledaements We wish t o thank J. E. Murphy and C . S . W i l l i a m s of the Central D a t a

Processing Department of the O a k Ridge Gaseous Diffusion P b t f o r i n % e r e s t b g discussions on numerical integration and f o r helpful suggestions.

References 1, D e J, Hughes, " P i l e Neutron Research", pp. 131-2, AddisonrlJesby

Fubl, Co., 2.

Cambridge, Mass.,

(1953)0

I. E, D q o n a.nd W, G, Pet%us, Nucleonics

3. C. H, Weetcott, J. Nuclear Energy 4.

3

a No.

12, po

86 (1957).

59 (1955).

D. J, Hughes and R e B. Schwartz, "Neutron Cross Sections", U. 8. A.l;aanics Energy ~aaranissionDocument BNL-325, second mition, (1958)

-27-

Appendix A

Wfective Cadmium Cutoffs Ec (in ev) for Spherical F i l t e r s o r Unidirectional Beams

12 EC

0253

10

15 20

30 40 80 100 120

.0300

10

15 20

30 40 50 60 80 100

120

.ob0

0.141 0.283 0.396 0.478 0.519 0.552 0.583 0 635 0.680 0 -719 0.158 0.285 0.395 0.477 0 . 5 ~ 0 555 0.584 0.635 0.679 0 -719

-

10

0.196

15

0.302

20

391 0.474 0.516

30 40 50 60 80 100 120

0

0

551

0.581 0.634 0.680 0.720

20

FRAm

0.459 0.163 0.082

0.067 0.068 0.071 0.073 0.074 0.455 0.162 0.083 0.068

Ec

0.126 0.235 0.375 0.477 0.518 0.552 0.583 0 e635 0.680 0 -719 0.144 0.245 0.374 0.476 0.520 0 * 555 0.584 0 9635 0 0679

0 *719

0.186 0.278 0 372 0.469 0.514 0 549 0 578 0 -633 0.680 0.720

30

m C T

0.585 0.241 0.0%

0.068

EC

0.118 0.202

0 * 351 0 475 f

ERACT

0.678 0.323 0.115 o .069

0.069

0.518 0 552 0.583 0 -635 Q .680 0 -719

0.581

0 J37

0.675

0.220

0 * 33-9 0

0,239 0.097

0.069

0 351 0.474 0.520 0.554 0.584 0 =635 0 -679 0 -719

0.181

0.265 0 * 357 0.462

0.511 0.547 0 578 0.632 0.679 Q -719

.UT

0.070

6280

Appendix A (Cont’d)

KFU.GT e

0500

/

15

0.230 0.3U

20

0,387

30

0 0457 0 .so2

10

40 50

60 80 100 120

.0600

10

0 0105

0,540 574 0.630 0.676 0.716

0.308 0,373 0 446 00493 0 532 0 566 0,625 0,672 0.714 0

0

‘9

0

331

30 40 60

0.376 0.1~32 0.467 0 536

100

0.650

20

15 20

30

40 60

.loo0

Ec 00222

01180 0.131

15

10

0,288 0 353 0 392 0 435

10

0 317 0 4 365 0

20

0.395

30

Oe49 0 477 0 526 0 597

40 60 100

0.206

0.529

0,623

15

0,286 0e 1 8 8 0.165

0 Q47L

100

30

20

12

0.245 0,2hl 0,202

Q.197

0.286 0.349 0,386 0 428 0. J+60 0.514 0,606 0 0 317 0 364 0 394 0 e 434 0 467 0 * 517 0 * $93

m m

Ec

’

m m

-29Appendix B Effective Boron Cutoffs Ec (in ev) for Spherical F i l t e r s or Unidirectional Beams

r = nv0/@, = Em

(4 0253

l0

(*./m2) 20

50 100 150 200 300 500

0350

20

50 100 150 200

300 500 .0500

20

50 100 150 200 300 500 *

0750

20

50

100 150 200

300 500 .loo0

20 50 100 150 200

300 500

12 EC

FRACT

0.384 0.545 0.0715 0.1672 0.524 0.5936 0.441 1.422 0.399 3.864 0.373 11.16 0.368 0.0353

0.0432 0.0858 0.1681 0.4192 1.139 3.615

0.356 0.538 0.571 0.492 0.430

E

C

20 FRAm

0.0347 0.0682 0.1258 0.2970 0.9990 0.377 3.600 0.368 11.13

0.407 0.611 0.655 0.543 0.449 0.383 0.368

0.367 0.569 0.638 0.563 0.472 0.393 0.369

0.0427

0.373

0.0832

0.586 0.682 0.627

0 -369

0.0541 0.1060 0.1879 0.3177 0.7782 3.000 10.92

0.324 0.517 0.603 0.561 0.485 0.404 0 371

0.0537 0.331 0.1046 0.538 0.1794 0.655 0.27~. 0.637 0.4997 0.553 2.586 0.429 10.74 0.373

0.2318

0.610

0.3389 0.621 0.5126 0.571 2.174 0.454 io .23 0.380 0.0840 0.1626 0.2728 0.3826 0.5230 1.483

9.155

0.263 0.455 0.599 0.641 0.623 0.510 0.395

me2

0.399 0.587 0.599 0.490 0.422

11.11

0.0699 0.288 0.1360 0.483

EC

0.0349 0.0692 0.1372 0.4227 1.207 3.743 11.14 0.0428 0.0841 0.1507 0.2781 0.8512 3.344 U. .06

0.382

30

0.0696 0.1348 0.2246 0.3168 0.4321

0.293

1.520

0.506

9.613

0.38

0.0837 0.1615 0.2685 0.3691 0.4796 0.9321 8.027

0.267 0.465 0.625 0.687 0.689 0.584 0.413

0.497 0.641 0.680 0.648

0.1442 0.2295 0.6227 3.046 11.00

0.0536 o.logg 0.~754 0.2531 0.3777 2.ll2 10.50

0.0694 0.1342 0.2223

0.3077 0.4030 1.049 8.925 0.0835 0.1610

0.2664 0.3629 0.4606 0.7403 6.881

0.518

0.406 0 * 370 0.336 0.550 0.685 0.690 0.618 0.458 0 - 376 0.296 0.505 0.662 0.716 0.703 0.561 0.398

0.269 0.471 0.639 o.‘p3 0.730 0.651 0.435

-30-

Appendix C Effective Cd Cutoffs Ec ( i n ev) for Cylinders (Hei&t/Diam.

= 2)

-

and Fractions of Reactions Below Cutoffs FRACT f o r Various Values of the Cnwacteristic Maxgellian Energy Em r = nv0/gr = 0

mils Cd

EC

12 FRAm

EC

30

20 FRAm

EC

FRACT

EC

FRACT

Em = 0.0253 10

15 20

30

40 60 100

,378 .419 .452 .504 -547 .615 .718

,0612

.0640 ,0661 .0684 -0705 .0730 .0756

-173 .339 ,431 .503 .547 -615 -718

0358 .116 .0751 ,0692 *0705 .0729 *0757 Em

10

15 20

30 40 60 100

.go6 .342 ,429 .5oi 0545 .615 .719

.148 .298 .419 .502 .547 .615 ,718 =

=

LO

15 20

30 40 60 100

.260 -353 .415 .483 -531 .606 .714

.301

01552 .12i8

.io03 -0915 .089

00799

.137 .259 .404 .501 .547 .615 .718

.466 .161 .0877 .0726 -0723 .0736 a760

.i8i .283 .401 0497 .544 .614 .718

.398

.245 .329 .390 .462 .513 .592 .7o8

.582 .218

.0907 .0698 .0708 .0728 -0755

0.035 .189 .308 .416 ,499 ,545 .614 .718

.346 ,1190 .0789 .0711 -07x5 .0734 .0760

482 .161 .0819 ,0695 .0706 .0730 .0756

.565 .213

.0997 a0750 .0733 .071+1 .0762

0.05 .250 ,338 .401 .472 ,522 -599 .712

.202

.153 .120

.io5 e0910

.0826

.479 .25’2 .188 .143 .122 .lo0

a60

-31(Appendix C continued) E,

.3U a375 .409 .454 .49i e555 .665

10

15 20

30 40 60 100

.336

234

.382

.2U

100

.bo3 .446

.203 *I79 .i40

10

30 40 60

.309

.yo

.210

15

.412 ,456 .495 .547 .631

0.075

.249 -175 .198

E,

20

=

,480

.540 .643

.304 .225 .248 .271 .266 .234

.177

.308 .366 *399 .441 .474 .529 .625

.344 .262

.289 .327 .326 .291 .219

= 0.100

.204 .197 .191 .225 ,236

.336 .381 ,410 .452 .488 ,539 .617

.267 .247 .244 .241 .23g *279 .304

Ec

mm

Ec

E~ = 0.70

#

-335

.go ,409 .450

.483 .534 .610

335 .270

,271 .q4 .274 -325 .370

r = nv0/fir = 20

mils Cd

EC

F!RAcT

Em = 0.20 10

15 20

30 40 60 100

mils

.37i .410 .439 .484 .520 '575 .657

.170 .185 a198 .220

.236 .259 .286

Em = 3.0

Ec

FFUiCT

# = 0.40 .390 .434 ,469 .520 .560 ,622 .7i5

-087 .lo2

.115 133 .145 .163 .192

Em = 10.0

Cd 10

15 20

30 40 60 100

,387 .429 .467 *537 .5g8

.0051 .0060 .0071

.702

.0181 00273

.859

.oog8

.0126

.5i7 .631 .716 .834 .9lO .995 1.073

,0031 .0049 .0063 .O083 .0095 .0104 ,0103

.403 .456 .494 .550 .591 .657 -757

.048

.061 .071 .082

.089

.loo

all9

.4ll .450 .485 .546 .596 .678 .792

FRACT = 1.0

.033 .036 .040 .Ob9 a057 .071 .OgO

632-

Appendix C (Cont'd)

Miscellaneous Results

r = nvo/15, = 12

mils Cd

EC

FRAm

E, 25

35 80

0.475 0.526 0.670

EC

m C T

EC

0,069

0,468 0.525 0,670

0,070

0.074

35

80

0.454

0.108

0.508

0,095 0.081

0.665

E25

35 80

0,436 0,476 0.592

m C T

0.432 0.488 0,656

0.072

0.070 0.074

0.473 0,524 0.670

OBL94 0.238

0.431 0.467 O.fl5

0,072 0.071 0,075

m m

0.035 0.464 0.523 0,670

O&Q 0,07lb

0,075

Em = 0.075

0.161 0.131 0.091

0.433 0,472 0,612

= U.100

0.200

EC

=

= 0,0253

E~ = 0.050 25

30

I2

30

0.274 0,274 0,361

,

0.208

0.422

0,316

0,207

0,458

0,529

0.157

0,579

0.252

Effective Boron cui;offs E~ (in ev) for cy1iniiers (ilei@;ht/biam. = 2)

and Fractions of Reactions belox Cutoffs FlRACT! far Various Values of the CharacterTstbc iyla;rweUau k r g y Em

n = 0.0253

E

-p__

20

422

30 50 80

, .Q40 *053 *0?9 0

100

0

125

440

.613 528

,618 --

173

623

150

0

039 053

0'77 QU5 .I49 535

9

1,631

200 300

.

5,160 15 292 611.227

500

1000

zn = 0,050 --1_1__

20

062

30

0082

50 80

9

100 150 200

3w 500 1000

061. e

082

061

a372

464

eo81

.EO 0

166

176

221

454

e

10297

0

D

4 ,620 15.126 61e 310

295 607

3 532 14.790 61 310

Appendix D (Continued)

r = nvo/fir

3.2 mg. B/m2

Ec

m C T

E~ 20

30

50

80

100 150 200 300

500 1000

106

324 410

154

o 522

.080

0

220

264 406 0770 34 532 14,518 61,272

30

20

Ec = 0,075 e

333-

.4639 540 420

607 ,628 627 ,600 0479 e 393* 371+

e 671 e679 .681 539 404 374 0

8

Em = 0,100 20

30

50

80 LOO 150 200

300 500

1000

0%

.E8 184 260 3O9 .442 638 2 570 1 3 e 494 61,189 0 0

298

.m .496 .594 628 645 645 539 .412 8

0

Q575

557 ~ 6 9 8 1 2 364 62,189

.620 43'9 376 0

ma

ORNL- 2823 Chemi.s t r y - General

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