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1

CH.VII: NEUTRON SLOWING DOWNINTRODUCTION

SLOWING DOWN VIA ELASTIC SCATTERING• KINEMATICS• SCATTERING LAW• LETHARGY• DIFFERENTIAL CROSS SECTIONS

SLOWING-DOWN EQUATION

• P1 APPROXIMATION • SLOWING-DOWN DENSITY• INFINITE HOMOGENEOUS MEDIA

SLOWING DOWN IN HYDROGEN

• HYPOTHESES• FLUX SHAPE• SLOWING-DOWN DENSITY SHAPE

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2

OTHER MODERATORS• PLACZEK FUNCTION • SYNTHETIC SLOWING-DOWN KERNELS

SPATIAL DEPENDENCE• FERMI’S AGE THEORY• SLOWING DOWN IN HYDROGEN

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3

VII.1 INTRODUCTION

Decrease of the n energy from Efission to Eth due to possibly both elastic and inelastic collisions

Inelastic collisions: E of the incident n > 1st excitation level of the nucleus

• 1st excited state for light nuclei: 1 MeV• 1st excited state for heavy nuclei: 0.1 MeV Inelastic collisions mainly with heavy nuclei… but for values

of E > resonance domain

Elastic collisions: not efficient with heavy nuclei With light nuclei (moderators)

Objective of this chapter: study of the n slowing down via elastic scattering with nuclei of mass A, in the resonance domain, to feed a multi-group diffusion model (see chap. IV) in groups of lower energy

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G

KINEMATICS

Absolute coordinates of n c.o.m. system

Velocity of the c.o.m. conserved Velocity modified only in direction in the c.o.m. system

G

Before collision After collision

Deflection angle:

4

VII.2 SLOWING DOWN VIA ELASTIC SCATTERING

Before collision After collision

E’ E

Deflection angle:

''' vv vv

''1

''

vA

Avv rr rrr vv '.o

rr '.

''1

1

v

AvG

nA

'vGv v

Gv

rv

'rv

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Minimum energy of a n after a collision

We have

Thus

Relations between variables

)'(1

'rrG A

A

vvvv

2

2

2

2

)1(

12

''

A

AA

v

v

E

E r

''1

12

min EEA

AE

)( ro f ').'(1

'1'.

1

ro A

A

v

vv

v )1(

'

1

1

rA

E

E

A

)(Efr A

A

E

E

A

Ar 2

1

'2

)1( 22

)(Efo E

EA

E

EAo

'

2

1

'2

1

)( or f )11(1 222 ooor AA

Element H 0

D2 0.111

C 0.716

U238 0.983

12

12

r

ro

AA

A

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6

SCATTERING LAW

= probability density function (pdf) of the deflection angle

Usually given in the c.o.m. system

Isotropic scattering (c.o.m.) :

In the lab system:

(cause vG small)

For A=1 :

Forward scattering only

Slowing-down kernel (i.e. pdf of the energy of the scattered n) – isotropic case

rrr ddp 2

1)(

oorr dpdp )()(

o

o

oo

A

A

Ap

2

1

21

2

1)(

22

22

2

11

A

00

02||)(

o

ooooo si

sip

rr dpdEEEK )()'|(

else

EEEifEEEK

0

''')1(

1)'|(

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Mean energy loss via elastic collision

with E’ with A because

LETHARGY

Eo : Eréf s.t. u>0 E Eo = 10 MeV

Elastic slowing-down kernel (isotropic scattering)

with

2

')1()'|()'('

'

'

EdEEEKEEEE

E

E

2)1(

2

2

)1(

A

AA (1-)/2

1 0.5

238 0.0083

E

Eu oln

dEEEKduuuK )'|()'|( due uu

1

)'(

1

ln'''' uuuEEE

q

1

ln

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Mean lethargy increment via elastic collision

Independent of u’!

As , =1 for A=1

Mean nb of collisions for a given lethargy increase: n s.t. u=n

Moderator quality

large + important scattering

Moderating power: s

Large moderation power + low absorption

Moderating ratio: s/a

1ln

11)'|()'('

1ln'

'

duuuKuuuu

u

u

1

1ln

2

)1(1

2

A

A

A

A

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9

u s.t. 2 MeV 1 eVa thermal

Moderator A n s s/a

H

D

H2O

D2O

C

U238

1

2

12

238

0

0.111

0.716

0.983

1

0.725

0.920

0.509

0.158

0.008

14

20

16

29

91

1730

1.35

0.176

0.060

0.003

71

5670

192

0.0092

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DIFFERENTIAL CROSS SECTIONS

Link between differential cross section and total scattering cross section slowing-down kernel

Differential cross section in lethargy and angle:

Cosinus of the deflection angle: determined by the elastic collision kinematics

Deflection angle determined by the lethargy increment!

)'|()',()',( uuKuruur ss

)(2

1)'|()',(),',',( oss fuuKuruur

E

EA

E

EAo

'

2

1

'2

1

2

'

2

'

2

1

2

1)'(

uuuu

o eA

eA

uu

))'(()( uuf ooo

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11

VII.3 SLOWING-DOWN EQUATION

P1 APPROXIMATION

Comments

Objective of the n slowing down: energy spectrum of the n in the domain of the elastic collisions

Input for multi-group diffusion

But no spatial variation of the flux no current no diffusion! Allowance to be given – even in a simple way – to the

spatial dependence

One speed case: with

Here with <o>0 (mainly if A1)

sottr tr

D

3

1

o

o

oo

A

A

Ap

2

1

21

2

1)(

22

22

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Steady-state Boltzmann equation in lethargy

(inelastic scattering accounted for in S (outside energy range))

Weak anisotropy

0th-order momentum 1st-order momentum (S isotropic)

)),(.3),((4

1),,( urJurur

),,('')',',(),',',(

),,(),(),,(

4

urSdduuruur

urururJdiv

s

u

o

t

),,('')',',()',()(4

1

')',()',(4

1),,(

4

urQdduururu

duuruururS

f

u

o

in

),(')',()',(

),(),(),(

urSduuruur

urururJdiv

s

u

o

t

0')',()',(

),(),(),(3

1

1

duurJuur

urJurur

s

u

o

t

duuruur ss '.),',',()',(1

dd

with

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For a mixture of isotopes:

Rem: energy domain of interest: resonance absorptions Elastic collisions only Inelastic scattering: fast domain impact on the source

SLOWING-DOWN DENSITY

Angular slowing-down density = nb of n (/volume.t) slowed down above lethargy u in a given point and direction:

Slowing-down density:

)'()'|()',()',(1 uuuuKuruur oiisii

s

)'|()',()',( uuKuruur isii

s

'')',',("),"',',(),,('

dduurduuururq su

u

o

')',(")"',(),( duurduuururq su

u

o

')',()',(")'|"( duururduuuK su

u

o

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Slowing-down current density:

Slowing-down density variation:

(interpretation?)

0th-order momentum:

with

')',()',(),(),(),(

duuruurururu

urqs

u

os

),(),(

),(),(),( urSu

urqurururJdiv ne

),(),(),(),(),( ururururur inastne

ddduurduuururq su

u

o'')',',("),"',',(),(

'1

')',(")"',(1 duurJduuursu

u

o

')',()',(")'|"( 1 duurJurduuuK su

u

o

),( uraresonance

domain

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15

Slowing-down current density variation

with

Slowing-down equations: summary

Outside the thermal and fast domains:

),(),(

),(),(),( urSu

urqurururJdiv ne

0),(

),(),(),(3

1 1

u

urqurJurur tr

')',()',(")'|"(),( duururduuuKurq su

u

o

')',()',(")'|"(),( 11 duurJurduuuKurq su

u

o

')',()',(),(),(),(

111 duurJuururJuru

urqs

u

os

0

),(),(),(),(

3

1 1

u

urqurJurur tr

),(),(),(),(),( 1 ururururur sioii

tsttr

1

2

3

4

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YS

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INFINITE HOMOGENEOUS MEDIA

Without spatial dependence:

Collision density:

Scattering probability with isotope i:

For an isotropic scattering:

with

Rem: F(u) and ci(u) smoother than t(u) and (u)

Slowing-down density:

Without absorption : for a source

q(E)/So = proba not to be absorbed between Esource and E = resonance escape proba if E = upper bound of thermal E

)(')'()'()()( uSduuuuuu s

u

ot

)(')'()'(1

1)( '

),0max(uSduuFuceuF i

uuu

quii i

)()()( uuuF t

)(

)()(

u

uuc

t

sii

iiq

1ln

)()()()(

uuuSdu

udqa

oSuuS )()( o

u

oSduuuq )()(

(interpretation?)

(units?)

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17

VII.4 SLOWING DOWN IN HYDROGEN

HYPOTHESES

Infinite media Absorption in H neglected Slowing down considered in the resonance domain Slowing down due to heavy nuclei neglected:

Elastic: minor contribution Inelastic: outside the energy range under study + low proportion of

heavy nuclei

FLUX SHAPE

)(')'()'()( ' uSduuFuceuF uuu

o )(

)()())(1(

)(uS

du

udSuFuc

du

udF

)0)0(( F')'()'()()(

""))(1(' duuSuceuSuF

duucu

o

u

u

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One speed source

for u > uo

Superposition of solutions of this type for a general S

Without absorption:

With absorption: Same behavior for (E) outside resonances (a negligible)

Reduction after each resonance by a factor

On the whole resonance domain, flux reduced by

)()( oo uuSuS

oot

osdu

u

u

Su

ueuF t

au

ou

)(

)()(

')'(

)'(

)()(

u

Su

t

o

EE

SE

t

o

)()(

')'(

)'(du

u

u

t

a

rese

')'(

)'('

)'(

)'( duu

udu

u

ut

a

resurest

au

ou ee

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19

SLOWING-DOWN DENSITY SHAPE

From the definition :

One speed source (uo) and u > uo

Resonance escape proba in u:

')'()'(")( "' duuFucdueuq uu

u

u

o

')'()'(' duuFuce uuu

o

)()( uSuF

oo

duuc

Suceuqu

ou )()('))'(1(

')'(

)'(exp

)(

)()( du

u

u

uq

uqup

t

au

uo

o

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20

VII.5 OTHER MODERATORS

Reminder: homogeneous media

PLACZEK FUNCTION

P(u) = collision density F(u) iff One material No absorption One speed source

with

)(')'()'(1

1)( '

),0max(uSduuFuceuF i

uuu

quii i

)(')'()'()( uduuPuuKuPo

)()(1

1)( uqHuHeuK u

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Laplace

Inverting term by term, effect of an increasing nb of collisions

Solution of ?

By intervals of width q

At the origin:

1st interval 0 < u < q :

Discontinuity in q :

2nd interval q < u < 2q :

)1)(1(

1)(

1

ppK

p

1)()()( pPpKpP

)(1

1)(

pKpP

...)()(1 2 pKpK

')'(1

)()('

),0max(duuP

euuP

uuu

qu

)()( uuP

1

)1

exp()(

u

uP

))(1

1(1

)exp()( 1

11 quuP

u

1

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Asymptotic behavior

Tauber’s theorem

Oscillations in the neighborhood of the origin =Placzek oscillations

1

)(1lim

)(lim

0

0

pK

p

pPpP

p

p

(1-

)P(u

)

u/q

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SYNTHETIC SLOWING-DOWN KERNELS

Integral slowing-down equation ordinary diff. eq. for H

diff. eq. with delay elseApproximations to simplify this diff. eq.

Wigner approximation

Asymptotic behavior of F(u) for an absorbing moderator, with c(u) cst, for a one speed S:

Approx. for a slow variation of c(u):

uc

as ec

uF

1

1)(

')'(

)'(

)(

)()(

duu

u

s

tas

t

au

oeu

uuF

(c1)

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Slowing-down density:

Asymptotic zone (Wigner):

Resonance escape proba in u

')'()'(")'|"()( duuFucduuuKuqu

u

o

')'()'("1

"''duuFucdu

e uuqu

u

u

qu

')'()'(1

'

duuFuce uuu

qu

)()('1

)('

uFucdue

uq as

uuu

quas

)()('

)'(

)'(

upeuqdu

u

u

ast

au

o

(u>q)

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Justification of the approximation

Mean nb of collisions to cross ui where c(ui) cst: ui/Proba to cross without absorption n consecutive intervals

ui/n :

Variation in the approximation

Outside the source domain:

Age approximation (see below)

Rem: compatible with for any c

)))(1(exp()))(1(1())(

)(.1( 1

iinn

iin

it

iai ucu

ucn

u

u

u

n

u

'))'(1(

1exp)))(1(exp()( duucuc

uup

u

oii

i

)()()(

uudu

udqa

as )()()(

)(uFuc

u

uas

s

a

)()(

)(uq

u

uas

s

a

uc

c

as ec

uF

1

1)(

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Greuling-Goertzel approximation

we consider

In the asymptotic zone

with

Yet

Thus

Resonance escape proba

))()(()'()()()'()'( uFucdu

duuuFucuFuc

))()(()()()( uFucdu

duFucuq

2

2

1u

du

udquFuc

)()()(

)()()(

uudu

udqa

)()(

)()(

uu

uqu

as

)()(

)()()(

uu

uqu

du

udq

as

a

')'()'(

)'(

)(du

uu

u

as

au

oeup

Rem: Wigner if Age if 0

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Generalization: synthetic kernels

Objective: replace the integral slowing-down eq.

by an ordinary differential eq. (i.e. without delay)

Synthetic kernel close to the initial kernel and s.t. approximated solution close enough to F(u)

Close? Momentums conservation:

Choice of the synthetic kernel? Solution of

approximated diff.eq. for the slowing-down density:

)(')'()'()'()( uSduuFucuuKuFo

)'(~

uuK )'( uuK

duuuKduuuKM k

o

k

ok )(~

)(

)'()()'(~)( uuuDuuKuL nm

)(~)()())()(

~)(( uFucuDuSuFuL nm

km

kkm du

duL

0

)(

kn

kkn du

duD

0

)( with

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Parameters of the differential operators Lm(u) and Dn(u)?

Conservation of m+n+1 momentums

1st-order synthetic kernels:

m=1, n=0 Wigner m=0, n=1 age m=1, n=1 Greuling – Goertzel

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29

VII.6 SPATIAL DEPENDENCE Slowing down in finite media

FERMI’S AGE THEORY

Use of the P1 equations with

• the age approximation:• neglected in the current equation

homogeneous zone, beyond the sources:

),()()(),( uruuurq s

u

urq

),(1

),(),(

),(),()),(),(( urSu

urqururururDdiv a

),(),(),(),(3

1),( ururDur

ururJ

tr

0),(

),()()(

),(),(

)()(

)(

u

urqurq

uu

ururq

uu

uD

s

a

s

and

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YS

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30

Let , with : n age [cm2] !!

Let

with the resonance escape proba

: slowing-down density without absorption

Equivalent to a time-dependent diffusion equation!

),(~)(),( rqprq

),(~ rq

')'(

1'

)'(

)'(2

)(

)(

d

Ldu

u

uo

s

au

o eep

')'()'(

)'()( du

uu

uDu

s

u

o

),(~

),(~rq

rq

0),(

),()(

1),(

2

rq

rqL

rqFermi’s equation

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YS

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31

Relation lethargy – time ?

Heavy nuclei mean lethargy increment low

low dispersion of the n lethargies same moderation

If slowing down identical for all n, u = f(slowing-down time) With all n with the same lethargy, the diffusion equation at time

t writes (for n emitted at t=0 with u=0):

Variation of u / u.t.:

Fermi’s equation

Approximation validity

Moderators heavy enough graphite in practice

vdtdu s

0),(),()),(),((),(1

trtrtrtrDdiv

t

tr

v a

PH

YS

-H4

06

– N

ucl

ea

r R

ea

cto

r P

hys

ics

– A

cad

em

ic y

ea

r 2

01

3-2

01

4

32

Examples of slowing-down kernels

Planar one speed source (Eo)

IC:

Point one speed source (Eo)

IC:

Mean square distance to the source:

Age = measure of the diffusion during the moderation

)()0,(~ oxxrq

)()0,(~ orrrq

2/3

4

||

)4(),(~

2

orr

erq

4),(~

4

|| 2oxx

erq

6

4),(~

4),(~

2

22

2

drrrq

drrrqrr

o

o

PH

YS

-H4

06

– N

ucl

ea

r R

ea

cto

r P

hys

ics

– A

cad

em

ic y

ea

r 2

01

3-2

01

4

33

Consistent age theory

Same treatment for as for

with

),(1 urq ),( urq

),()()(),( 11 urJuuurq s

'")'"(11 duduuuKu

u

o

)()()(1 uuKuK o

22

2

1

2

1)(

uu

o eA

eA

u